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One of the basic concepts in mathematics. Let two sets $X$ and $Y$ be given and suppose that to each element $x\in X$ corresponds an element $y\in Y$, which is denoted by $f(x)$. In this case one says that a function $f$ is given on $X$ (and also that the variable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$.
 
One of the basic concepts in mathematics. Let two sets $X$ and $Y$ be given and suppose that to each element $x\in X$ corresponds an element $y\in Y$, which is denoted by $f(x)$. In this case one says that a function $f$ is given on $X$ (and also that the variable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$.
  
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The concept of a function considered below is based on the concept of a set and of the simplest operations on sets.
 
The concept of a function considered below is based on the concept of a set and of the simplest operations on sets.
  
One says that the number of elements of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194013.png" /> is equal to 1 or that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194014.png" /> consists of one element if it contains an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194015.png" /> and no others (in other words, if after deleting the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194016.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194017.png" /> one obtains the empty set). A non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194018.png" /> is called a set with two elements, or a pair, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194019.png" />, if after deleting a set consisting of only one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194020.png" /> there remains a set also consisting of one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194021.png" /> (this definition does not depend on the choice of the chosen element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194022.png" />).
+
One says that the number of elements of a set $  A $
 +
is equal to 1 or that the set $  A $
 +
consists of one element if it contains an element $  a $
 +
and no others (in other words, if after deleting the set $  \{ a \} $
 +
from $  A $
 +
one obtains the empty set). A non-empty set $  A $
 +
is called a set with two elements, or a pair, $  A = \{ a,\  b \} $,  
 +
if after deleting a set consisting of only one element $  a \in A $
 +
there remains a set also consisting of one element $  b \in A $(
 +
this definition does not depend on the choice of the chosen element $  a \in A $).
  
If a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194023.png" /> is given, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194024.png" /> is called the ordered pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194026.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194027.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194028.png" /> is called its first element and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194029.png" /> is called the second element.
+
If a pair $  A = \{ a,\  b \} $
 +
is given, then the pair $  \{ a,\  \{ a,\  b \} \} $
 +
is called the ordered pair of elements $  a \in A $
 +
and $  b \in A $
 +
and is denoted by $  (a,\  b) $.  
 +
The element $  a \in A $
 +
is called its first element and $  b \in A $
 +
is called the second element.
  
Given sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194031.png" />, the set of all ordered pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194034.png" />, is called the product of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194036.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194037.png" />. It is not assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194038.png" /> is different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194039.png" />, that is, it is possible that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194040.png" />.
+
Given sets $  X $
 +
and $  Y $,  
 +
the set of all ordered pairs $  (x,\  y) $,  
 +
$  x \in X $,  
 +
$  y \in Y $,  
 +
is called the product of the sets $  X $
 +
and $  Y $
 +
and is denoted by $  X \times Y $.  
 +
It is not assumed that $  X $
 +
is different from $  Y $,  
 +
that is, it is possible that $  X = Y $.
  
Each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194041.png" /> of ordered pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194044.png" />, such that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194047.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194048.png" />, is called a function or, what is the same thing, a mapping.
+
Each set $  f = \{ (x,\  y) \} $
 +
of ordered pairs $  (x,\  y) $,  
 +
$  x \in X $,  
 +
$  y \in Y $,  
 +
such that, if $  (x ^ \prime  ,\  y ^ \prime  ) \in f $
 +
and $  (x ^{\prime\prime} ,\  y ^{\prime\prime} ) \in f $,  
 +
then $  y ^ \prime  \neq y ^{\prime\prime} $
 +
implies that $  x ^ \prime  \neq x ^{\prime\prime} $,  
 +
is called a function or, what is the same thing, a mapping.
  
 
As well as the terms  "function"  and  "mapping"  one uses in certain situations the terms  [[transformation]] ,  [[morphism]] ,  [[correspondence]] , which are equivalent to them.
 
As well as the terms  "function"  and  "mapping"  one uses in certain situations the terms  [[transformation]] ,  [[morphism]] ,  [[correspondence]] , which are equivalent to them.
  
The set of all first elements of ordered pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194049.png" /> of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194050.png" /> is called the [[Domain of definition|domain of definition]] (or the set of definition) of this function and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194051.png" />, and the set of all second elements is called the [[Range of values|range of values]] (the set of values) and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194052.png" />. The set of ordered pairs itself, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194053.png" />, considered as a subset of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194054.png" />, is called the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194055.png" />.
+
The set of all first elements of ordered pairs $  (x,\  y) $
 +
of a given function f $
 +
is called the [[Domain of definition|domain of definition]] (or the set of definition) of this function and is denoted by $  X _{f} $,  
 +
and the set of all second elements is called the [[Range of values|range of values]] (the set of values) and is denoted by $  Y _{f} $.  
 +
The set of ordered pairs itself, $  f = \{ (x,\  y) \} $,  
 +
considered as a subset of the product $  X \times Y $,  
 +
is called the graph of f $.
  
The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194056.png" /> is called the argument of the function, or the independent variable, and the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194057.png" /> is called the dependent variable.
+
The element $  x \in X $
 +
is called the argument of the function, or the independent variable, and the element $  y \in Y $
 +
is called the dependent variable.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194058.png" /> is a function, then one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194059.png" /> and says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194060.png" /> maps the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194061.png" /> into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194062.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194063.png" /> one simply writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194064.png" />.
+
If $  f = \{ (x,\  y) \} $
 +
is a function, then one writes $  f: \  X _{f} \rightarrow Y $
 +
and says that f $
 +
maps the set $  X _{f} $
 +
into the set $  Y $.  
 +
In the case $  X = X _{f} $
 +
one simply writes $  f: \  X \rightarrow Y $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194065.png" /> is a function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194066.png" />, then one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194067.png" /> (sometimes simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194068.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194069.png" />) and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194072.png" />, and says that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194073.png" /> puts the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194074.png" /> in correspondence with the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194075.png" /> (the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194076.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194077.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194078.png" />) or, what is the same, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194079.png" /> corresponds to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194080.png" />. In this case one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194081.png" /> is the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194082.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194083.png" />, or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194084.png" /> is the image of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194085.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194086.png" />.
+
If $  f: \  X \rightarrow Y $
 +
is a function and $  (x,\  y) \in f $,  
 +
then one writes $  y = f (x) $(
 +
sometimes simply $  y = fx $
 +
or $  y = x f \  $)  
 +
and also $  f: \  x \mapsto y $,  
 +
$  x \in X $,  
 +
$  y \in Y $,  
 +
and says that the function f $
 +
puts the element $  y $
 +
in correspondence with the element $  x $(
 +
the mapping f $
 +
maps $  x $
 +
to $  y $)  
 +
or, what is the same, the element $  y $
 +
corresponds to the element $  x $.  
 +
In this case one also says that $  y $
 +
is the value of f $
 +
at the point $  x $,  
 +
or that $  y $
 +
is the image of the element $  x $
 +
under f $.
  
As well as the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194087.png" /> one also uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194088.png" /> for denoting the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194089.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194090.png" />.
+
As well as the symbol $  f (x _{0} ) $
 +
one also uses the notation $  f (x) \mid  _{ {x = x _ 0}} $
 +
for denoting the value of f $
 +
at $  x _{0} $.
  
Sometimes the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194091.png" /> itself is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194092.png" />. Denoting both the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194093.png" /> and its value at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194094.png" /> by the same symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194095.png" /> does not usually lead to misunderstanding, since in any particular case, as a rule, it is always clear what one is talking about. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194096.png" /> often turns out to be more convenient than the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194097.png" /> in computations. For example, writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194098.png" /> is more convenient and simpler to use in analytic manipulations than writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f04194099.png" />.
+
Sometimes the function f $
 +
itself is denoted by the symbol f (x) $.  
 +
Denoting both the function $  f: \  X \rightarrow Y $
 +
and its value at the point $  x \in X $
 +
by the same symbol f (x) $
 +
does not usually lead to misunderstanding, since in any particular case, as a rule, it is always clear what one is talking about. The notation f (x) $
 +
often turns out to be more convenient than the notation $  f: \  x \mapsto y $
 +
in computations. For example, writing $  f (x) = x ^{2} $
 +
is more convenient and simpler to use in analytic manipulations than writing $  f: \  x \mapsto x ^{2} $.
  
Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940100.png" />, the set of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940101.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940102.png" /> is called the pre-image of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940103.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940104.png" />. Thus,
+
Given $  y \in Y $,  
 +
the set of all elements $  x \in X $
 +
such that $  f (x) = y $
 +
is called the pre-image of the element $  y $
 +
and is denoted by f ^ {\  -1} (y) $.  
 +
Thus,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940105.png" /></td> </tr></table>
+
$$
 +
f ^ {\  -1} (y) \  = \  \{ {x} : {x \in X,\  f (x) = y} \}
 +
.
 +
$$
  
Obviously, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940106.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940107.png" />, the empty set.
+
Obviously, if $  y \in Y \setminus Y _{f} $,  
 +
then $  f ^ {\  -1} (y) = \emptyset $,  
 +
the empty set.
  
Let a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940108.png" /> be given. In other words, to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940109.png" /> corresponds a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940110.png" /> and to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940111.png" /> corresponds at least one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940112.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940113.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940114.png" /> maps the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940115.png" /> into itself. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940116.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940117.png" /> coincides with the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940118.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940119.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940120.png" /> onto the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940121.png" /> or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940122.png" /> is a surjective mapping, or, more concisely, it is a [[Surjection|surjection]]. Thus, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940123.png" /> is a surjection if for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940124.png" /> there is at least one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940125.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940126.png" />.
+
Let a mapping $  f: \  X \rightarrow Y $
 +
be given. In other words, to each $  x \in X $
 +
corresponds a unique element $  y \in Y $
 +
and to each $  y \in Y _{f} \subset Y $
 +
corresponds at least one element $  x \in X $.  
 +
If $  Y = X $,  
 +
one says that f $
 +
maps the set $  X $
 +
into itself. If $  Y = Y _{f} $,  
 +
that is, if $  Y $
 +
coincides with the range of f $,  
 +
then one says that f $
 +
maps $  X $
 +
onto the set $  Y $
 +
or that f $
 +
is a surjective mapping, or, more concisely, it is a [[Surjection|surjection]]. Thus, a mapping $  f: \  X \rightarrow Y $
 +
is a surjection if for each element $  y \in Y $
 +
there is at least one element $  x \in X $
 +
such that $  f (x) = y $.
  
If under a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940127.png" /> different elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940128.png" /> correspond to different elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940129.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940130.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940131.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940132.png" /> is said to be a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940133.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940134.png" /> and also a univalent mapping or an [[Injection|injection]]. Thus, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940135.png" /> is univalent (injective) if and only if the pre-image of each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940136.png" /> belonging to the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940137.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940138.png" />, consists precisely of one element. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940139.png" /> is simultaneously one-to-one and onto the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940140.png" /> (see [[One-to-one correspondence|One-to-one correspondence]]), that is, is at the same time injective and surjective, then it is called a bijective mapping or a [[Bijection|bijection]].
+
If under a mapping $  f: \  X \rightarrow Y $
 +
different elements $  y \in Y $
 +
correspond to different elements $  x \in X $,  
 +
that is, if $  x ^ \prime  \neq x ^{\prime\prime} $
 +
implies  $  f (x ^ \prime  ) \neq f (x ^{\prime\prime} ) $,  
 +
then f $
 +
is said to be a one-to-one mapping of $  X $
 +
into $  Y $
 +
and also a univalent mapping or an [[Injection|injection]]. Thus, a mapping $  f: \  X \rightarrow Y $
 +
is univalent (injective) if and only if the pre-image of each element $  y $
 +
belonging to the range of f $,  
 +
that is, $  y \in Y _{f} $,  
 +
consists precisely of one element. If the mapping $  f: \  X \rightarrow Y $
 +
is simultaneously one-to-one and onto the set $  Y $(
 +
see [[One-to-one correspondence|One-to-one correspondence]]), that is, is at the same time injective and surjective, then it is called a bijective mapping or a [[Bijection|bijection]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940142.png" />, then the set
+
If $  f: \  X \rightarrow Y $
 +
and $  A \subset X $,  
 +
then the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940143.png" /></td> </tr></table>
+
$$
 +
S \  = \  \{ {y} : {y \in Y,\
 +
y = f (x),\  x \in A} \}
 +
,
 +
$$
  
that is, the set of all those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940144.png" /> such that there is at least one element of the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940145.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940146.png" /> which is mapped to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940147.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940148.png" />, is called the image of the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940149.png" />, and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940150.png" />. In particular, always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940151.png" />. The following relations are true for the images of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940153.png" />:
+
that is, the set of all those $  y $
 +
such that there is at least one element of the subset $  A $
 +
of $  X $
 +
which is mapped to $  y $
 +
by f $,  
 +
is called the image of the subset $  A $,  
 +
and one writes $  S = f (A) $.  
 +
In particular, always $  Y _{f} = f (X) $.  
 +
The following relations are true for the images of sets $  A \subset X $
 +
and $  B \subset X $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940154.png" /></td> </tr></table>
+
$$
 +
f (A \cup B) \  = \
 +
f (A) \cup f (B),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940155.png" /></td> </tr></table>
+
$$
 +
f (A \cap B) \  \subset \  f (A) \cap f (B),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940156.png" /></td> </tr></table>
+
$$
 +
f (A) \setminus f (B) \  \subset \  f (A \setminus B),
 +
$$
  
and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940157.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940158.png" />.
+
and if $  A \subset B $,  
 +
then $  f (A) \subset f (B) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940159.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940160.png" />, then the set
+
If $  f: \  X \rightarrow Y $
 +
and $  S \subset Y $,  
 +
then the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940161.png" /></td> </tr></table>
+
$$
 +
A \  = \  \{ {x} : {x \in X,\
 +
f (x) \in S} \}
 +
$$
  
is called the pre-image of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940162.png" /> and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940163.png" />. Thus, the pre-image of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940164.png" /> consists of all those elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940165.png" /> which are mapped to elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940166.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940167.png" />, or what is the same thing, it consists of all pre-images of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940168.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940169.png" />. For the pre-images of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940170.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940171.png" /> the relations
+
is called the pre-image of the set $  S $
 +
and one writes $  A = f ^ {\  -1} (S) $.  
 +
Thus, the pre-image of a set $  S $
 +
consists of all those elements $  x \in X $
 +
which are mapped to elements of $  S $
 +
under f $,  
 +
or what is the same thing, it consists of all pre-images of elements $  y \in S $:  
 +
f ^ {\  -1} (S) = \cup _{ {y \in S}} f ^ {\  -1} (y) $.  
 +
For the pre-images of sets $  S \subset Y $
 +
and $  T \subset Y $
 +
the relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940172.png" /></td> </tr></table>
+
$$
 +
f ^ {\  -1} (S \cup T) \  = \
 +
f ^ {\  -1} (S) \cup f ^ {\  -1} (T),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940173.png" /></td> </tr></table>
+
$$
 +
f ^ {\  -1} (S \cap T) \  = \  f ^ {\  -1} (S) \cap f ^ {\  -1} (T),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940174.png" /></td> </tr></table>
+
$$
 +
f ^ {\  -1} (S \setminus T) \  = \  f ^ {\  -1} (S) \setminus f ^ {\  -1} (T)
 +
$$
  
are true, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940175.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940176.png" />.
+
are true, and if $  S \subset T $,  
 +
then $  f ^ {\  -1} (S) \subset f ^ {\  -1} (T) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940177.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940178.png" /> generates in a natural way a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940179.png" /> under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940180.png" /> corresponds to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940181.png" />. This function is called the restriction of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940182.png" /> to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940183.png" /> and is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940184.png" />. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940185.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940186.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940187.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940188.png" /> does not coincide with the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940189.png" />, then the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940190.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940191.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940192.png" /> may have a different domain of definition than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940193.png" /> and, consequently, is different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940194.png" />.
+
If $  A \subset X $,  
 +
then the function $  f: \  X \rightarrow Y $
 +
generates in a natural way a function defined on $  A $
 +
under which f (x) $
 +
corresponds to the element $  x \in A $.  
 +
This function is called the restriction of the function f $
 +
to the set $  A $
 +
and is sometimes denoted by f _{A} $.  
 +
Thus $  f _{A} : \  A \rightarrow Y $
 +
and for any $  x \in A $
 +
one has $  f _{A} : \  x \mapsto f (x) $.  
 +
If the set $  A $
 +
does not coincide with the set $  X $,  
 +
then the restriction f _{A} $
 +
of f $
 +
to $  A $
 +
may have a different domain of definition than f $
 +
and, consequently, is different from f $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940195.png" />, if each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940196.png" /> is a certain set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940197.png" />, and if, moreover, among these sets there is at least one set consisting of more than one element, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940198.png" /> is called a [[Multi-valued function|multi-valued function]] (sometimes, many-valued function). Further, the elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940199.png" /> are often called the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940200.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940201.png" />. If each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940203.png" />, consists of only one element, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940204.png" /> is also called a single-valued function.
+
If $  f: \  X \rightarrow Y $,  
 +
if each element $  y \in Y _{f} $
 +
is a certain set of elements $  y = \{ z \} $,  
 +
and if, moreover, among these sets there is at least one set consisting of more than one element, then f $
 +
is called a [[Multi-valued function|multi-valued function]] (sometimes, many-valued function). Further, the elements of the set $  f (x) = \{ z \} $
 +
are often called the values of f $
 +
at $  x $.  
 +
If each set f (x) $,  
 +
$  x \in X $,  
 +
consists of only one element, then the function f $
 +
is also called a single-valued function.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940206.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940207.png" /> defined for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940208.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940209.png" /> is called the composition (superposition) of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940211.png" />, also the [[Composite function|composite function]], and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940212.png" />.
+
If $  f: \  X \rightarrow Y $
 +
and $  g: \  Y \rightarrow Z $,  
 +
then the function $  F: \  X \rightarrow Z $
 +
defined for each $  x \in X $
 +
by the formula $  F (x) = g (f (x)) $
 +
is called the composition (superposition) of the functions f $
 +
and $  g $,  
 +
also the [[Composite function|composite function]], and is denoted by $  g \circ f $.
  
Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940213.png" /> be given and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940214.png" /> be its range. The set of all possible ordered pairs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940215.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940216.png" />, forms a function, called the [[Inverse function|inverse function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940217.png" /> and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940218.png" />. Under the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940219.png" />, to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940220.png" /> corresponds the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940221.png" />, that is, a certain set of elements. So the inverse function is, generally speaking, a multi-valued function. If a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940222.png" /> is injective, then the inverse mapping is a single-valued function and maps the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940223.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940224.png" /> onto the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940225.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940226.png" />.
+
Let a function $  f: \  X \rightarrow Y $
 +
be given and let $  Y _{f} $
 +
be its range. The set of all possible ordered pairs of the form $  (y,\  f ^ {\  -1} (y)) $,  
 +
$  y \in Y _{f} $,  
 +
forms a function, called the [[Inverse function|inverse function]] of f $
 +
and denoted by f ^ {\  -1} $.  
 +
Under the inverse function f ^ {\  -1} $,  
 +
to each $  y \in Y _{f} $
 +
corresponds the pre-image f ^ {\  -1} (y) $,  
 +
that is, a certain set of elements. So the inverse function is, generally speaking, a multi-valued function. If a mapping $  f: \  X \rightarrow Y $
 +
is injective, then the inverse mapping is a single-valued function and maps the range $  Y _{f} $
 +
of f $
 +
onto the domain of definition $  X $
 +
of f $.
  
 
==Functions on numbers.==
 
==Functions on numbers.==
An important class of functions is that of the complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940227.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940228.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940229.png" /> is the set of all complex numbers. One can carry out various arithmetical operations on complex-valued functions. If two given complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940230.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940231.png" /> are defined on the same set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940232.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940233.png" /> is a complex number, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940234.png" /> is defined as the function taking the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940235.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940236.png" />; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940237.png" /> is the function taking the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940238.png" /> at each point; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940239.png" /> is the function taking the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940240.png" /> at each point; and, finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940241.png" /> is the function equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940242.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940243.png" /> (which, of course, makes sense only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940244.png" />).
+
An important class of functions is that of the complex-valued functions $  f: \  X \rightarrow Y $,  
 +
$  Y \subset \mathbf C $,  
 +
where $  \mathbf C $
 +
is the set of all complex numbers. One can carry out various arithmetical operations on complex-valued functions. If two given complex-valued functions f $
 +
and $  g $
 +
are defined on the same set $  X $
 +
and if $  \lambda $
 +
is a complex number, then the function $  \lambda f $
 +
is defined as the function taking the value $  \lambda f (x) $
 +
at each point $  x \in X $;  
 +
the function f + g $
 +
is the function taking the value f (x) + g (x) $
 +
at each point; the function $  fg $
 +
is the function taking the value f (x) g (x) $
 +
at each point; and, finally, f/g $
 +
is the function equal to f (x)/g (x) $
 +
at each point $  x \in X $(
 +
which, of course, makes sense only when $  g (x) \neq 0 $).
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940245.png" /> is called a real-valued function (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940246.png" /> is the set of real numbers). A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940247.png" /> is said to be bounded from above (bounded from below) on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940248.png" /> if its range is bounded from above (bounded from below). In other words, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940249.png" /> is bounded from above (bounded from below) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940250.png" /> if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940251.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940252.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940253.png" /> is satisfied (the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940254.png" /> is satisfied, respectively). A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940255.png" /> that is both bounded from above and from below on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940256.png" /> is simply said to be bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940257.png" />. An upper (lower) bound of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940258.png" /> is an upper (lower) bound of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940259.png" />.
+
A function $  f: \  X \rightarrow \mathbf R $
 +
is called a real-valued function ( $  \mathbf R $
 +
is the set of real numbers). A real-valued function $  f: \  X \rightarrow \mathbf R $
 +
is said to be bounded from above (bounded from below) on the set $  X $
 +
if its range is bounded from above (bounded from below). In other words, a function $  f: \  X \rightarrow \mathbf R $
 +
is bounded from above (bounded from below) on $  X $
 +
if there is a constant $  c \in \mathbf R $
 +
such that for every $  x \in X $
 +
the inequality f (x) \leq c $
 +
is satisfied (the inequality f (x) \geq c $
 +
is satisfied, respectively). A function f $
 +
that is both bounded from above and from below on $  X $
 +
is simply said to be bounded on $  X $.  
 +
An upper (lower) bound of the range of $  f: \  X \rightarrow \mathbf R $
 +
is an upper (lower) bound of the function f $.
  
A major role in mathematical analysis is played by functions on numbers or, more precisely, complex-valued functions of a complex variable, that is, functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940260.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940261.png" />. If the domain of definition of such a function and its range are both subsets of the real numbers, then this function is called a real function, or, more precisely, a real-valued function of a real variable. The generalization of the concept of a function on numbers is, first of all, a complex-valued function of several complex variables, called a complex function of several variables. A further generalization of a function on numbers is a vector-valued function (see [[Vector function|Vector function]]) and, in general, a function for which the domain of definition and the range are provided with definite structures. For example, if the ranges of functions belong to a certain vector space, then such functions can be added; if they belong to a ring, then the functions can be added and multiplied; if they belong to a set which is ordered in some specific way, then one can generalize to these functions the idea of boundedness, upper and lower bounds, etc. The presence of topological structures on the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940262.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940263.png" /> enables one to introduce the concept of a [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940264.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940265.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940266.png" /> are topological vector spaces one introduces the concept of differentiability for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940267.png" /> (cf. [[Differentiable function|Differentiable function]]).
+
A major role in mathematical analysis is played by functions on numbers or, more precisely, complex-valued functions of a complex variable, that is, functions $  f: \  X \rightarrow Y $,  
 +
where $  X,\  Y \subset \mathbf C $.  
 +
If the domain of definition of such a function and its range are both subsets of the real numbers, then this function is called a real function, or, more precisely, a real-valued function of a real variable. The generalization of the concept of a function on numbers is, first of all, a complex-valued function of several complex variables, called a complex function of several variables. A further generalization of a function on numbers is a vector-valued function (see [[Vector function|Vector function]]) and, in general, a function for which the domain of definition and the range are provided with definite structures. For example, if the ranges of functions belong to a certain vector space, then such functions can be added; if they belong to a ring, then the functions can be added and multiplied; if they belong to a set which is ordered in some specific way, then one can generalize to these functions the idea of boundedness, upper and lower bounds, etc. The presence of topological structures on the sets $  X $
 +
and $  Y $
 +
enables one to introduce the concept of a [[Continuous function|continuous function]] $  f: \  X \rightarrow Y $.  
 +
In the case when $  X $
 +
and $  Y $
 +
are topological vector spaces one introduces the concept of differentiability for a function $  f: \  X \rightarrow Y $(
 +
cf. [[Differentiable function|Differentiable function]]).
  
 
==Methods for specifying functions.==
 
==Methods for specifying functions.==
 
Functions on numbers (and certain generalizations of them) can be given by formulas. This is the analytic method for specifying functions. For this one uses a certain supply of functions which have been studied and have a special notation (primarily the elementary functions), algebraic operations, composition of functions and limit transitions (which includes operations of mathematical analysis such as differentiation, integration, summing series), for example:
 
Functions on numbers (and certain generalizations of them) can be given by formulas. This is the analytic method for specifying functions. For this one uses a certain supply of functions which have been studied and have a special notation (primarily the elementary functions), algebraic operations, composition of functions and limit transitions (which includes operations of mathematical analysis such as differentiation, integration, summing series), for example:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940268.png" /></td> </tr></table>
+
$$
 +
y \  = \  ax + b,\ \
 +
y \  = \  ax ^{2} ,\ \
 +
y \  = \  1 + \sqrt { \mathop{\rm log}\nolimits \  \cos \  2 \pi x} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940269.png" /></td> </tr></table>
+
$$
 +
\zeta (z) \  = \  \sum _ {n = 1} ^ \infty {
 +
\frac{1}{n ^ z}
 +
} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940270.png" /></td> </tr></table>
+
$$
 +
P _{n} (x) \  = \  {
 +
\frac{1}{2 ^{n} n!}
 +
}
 +
\frac{d ^{n} (x ^{2} - 1) ^ n}{dx ^ n}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940271.png" /></td> </tr></table>
+
$$
 +
I ( \alpha ,\  \beta ) \  = \  \int\limits _{0} ^ {+ \infty}
 +
e ^ {- \alpha x} {
 +
\frac{\sin \  \beta x}{x}
 +
} \  dx.
 +
$$
  
The class of functions that are presented, in a well-determined sense, as the sum of series, even only as sums of trigonometric series, is very wide. A function can be given analytically either in an explicit form, that is, by a formula of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940272.png" />, or as an [[Implicit function|implicit function]], that is, by an equation of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940273.png" />. Sometimes the function is given with the aid of several formulas, for example,
+
The class of functions that are presented, in a well-determined sense, as the sum of series, even only as sums of trigonometric series, is very wide. A function can be given analytically either in an explicit form, that is, by a formula of the type $  y = f (x) $,  
 +
or as an [[Implicit function|implicit function]], that is, by an equation of the type $  F (x,\  y) = 0 $.  
 +
Sometimes the function is given with the aid of several formulas, for example,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940274.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{*}
 +
f (x) \  = \  \left \{
  
A function can also be given by using a description of the correspondence. Let, for example, the number 1 correspond to every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940275.png" />, the number 0 to the number 0, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940276.png" /> to every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940277.png" />. As a result one obtains a function defined on the whole real line and taking three values: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940278.png" />. This function has the special notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940279.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940280.png" />). Another example: the number 1 corresponds to each rational number and the number 0 to each irrational number. The function obtained is called the [[Dirichlet-function|Dirichlet function]]. The same function can be given in different ways; for example, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940281.png" /> and the Dirichlet function can be defined not only by verbal descriptions but also by formulas.
+
\begin{array}{ll}
 +
2 ^{x} \  &\textrm{ if } \  x > 0,  \\
 +
0 \  &\textrm{ if } \  x = 0,  \\
 +
x - 1 \  &\textrm{ if } \  x < 0.  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
A function can also be given by using a description of the correspondence. Let, for example, the number 1 correspond to every $  x > 0 $,  
 +
the number 0 to the number 0, and the number $  -1 $
 +
to every $  x < 0 $.  
 +
As a result one obtains a function defined on the whole real line and taking three values: $  1,\  0,\  -1 $.  
 +
This function has the special notation $  \mathop{\rm sign}\nolimits \  x $(
 +
or $  \mathop{\rm sgn}\nolimits \  x $).  
 +
Another example: the number 1 corresponds to each rational number and the number 0 to each irrational number. The function obtained is called the [[Dirichlet-function|Dirichlet function]]. The same function can be given in different ways; for example, the function $  \mathop{\rm sign}\nolimits \  x $
 +
and the Dirichlet function can be defined not only by verbal descriptions but also by formulas.
  
 
Every formula is a symbolic notation of a certain previously-described correspondence, so that in the end there is no fundamental difference between specifying a function by a formula or by a verbal description of the correspondence; this difference is superficial. It should be borne in mind that every function newly defined by some means or other can, if a special notation is introduced for it, serve to define other functions by using formulas including this new symbol. However, for an analytic representation of a function the supply of functions and operations that are to be used in the formulas is very essential; usually one tries to make this supply as small as possible and chooses the functions and operations as simply as possible in a certain well-determined sense.
 
Every formula is a symbolic notation of a certain previously-described correspondence, so that in the end there is no fundamental difference between specifying a function by a formula or by a verbal description of the correspondence; this difference is superficial. It should be borne in mind that every function newly defined by some means or other can, if a special notation is introduced for it, serve to define other functions by using formulas including this new symbol. However, for an analytic representation of a function the supply of functions and operations that are to be used in the formulas is very essential; usually one tries to make this supply as small as possible and chooses the functions and operations as simply as possible in a certain well-determined sense.
  
When one is concerned with real-valued functions of a single real variable, then to give an intuitive representation of the nature of the functional dependence one often constructs the graph of the function in the coordinate plane, in other words, given a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940282.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940283.png" />, one considers the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940284.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940285.png" />, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940286.png" />-plane. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940287.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940288.png" />
+
When one is concerned with real-valued functions of a single real variable, then to give an intuitive representation of the nature of the functional dependence one often constructs the graph of the function in the coordinate plane, in other words, given a function $  f: \  X \rightarrow \mathbf R $,  
 +
$  X \subset \mathbf R $,  
 +
one considers the set of points $  (x,\  f (x)) $,  
 +
$  x \in X $,  
 +
in the $  (x,\  y) $-
 +
plane. $  y = x - 1 $
 +
$  y = 2 ^{x} $
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041940a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041940a.gif" />
Line 111: Line 410:
 
Figure: f041940b
 
Figure: f041940b
  
Thus, the graph of the function (*) has the form depicted in Fig. a, the graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940289.png" /> is Fig. b, and the graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940290.png" /> consists of the isolated points in Fig. c.
+
Thus, the graph of the function (*) has the form depicted in Fig. a, the graph of the function $  \mathop{\rm sign}\nolimits \  x $
 +
is Fig. b, and the graph of the function $  y = 1 + \sqrt { \mathop{\rm log}\nolimits \  \cos \  2 \pi x} $
 +
consists of the isolated points in Fig. c.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041940c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041940c.gif" />
Line 124: Line 425:
 
The simplest complex functions are the [[Elementary functions|elementary functions]], among which one distinguishes the algebraic polynomials, the trigonometric polynomials and also the rational functions (cf. [[Rational function|Rational function]]). The special role of these functions is that one of the methods for studying and using more general functions is based on approximating them by algebraic polynomials, by trigonometric polynomials or by rational functions, as well as by functions composed from these functions in some well-determined way (see [[Spline|Spline]]). The branch of the theory of functions that studies the approximation of functions by collections of functions that are simple in a certain sense is called [[Approximation theory|approximation theory]]. In this theory approximations of functions by linear aggregates of eigen functions of certain operators are also very important.
 
The simplest complex functions are the [[Elementary functions|elementary functions]], among which one distinguishes the algebraic polynomials, the trigonometric polynomials and also the rational functions (cf. [[Rational function|Rational function]]). The special role of these functions is that one of the methods for studying and using more general functions is based on approximating them by algebraic polynomials, by trigonometric polynomials or by rational functions, as well as by functions composed from these functions in some well-determined way (see [[Spline|Spline]]). The branch of the theory of functions that studies the approximation of functions by collections of functions that are simple in a certain sense is called [[Approximation theory|approximation theory]]. In this theory approximations of functions by linear aggregates of eigen functions of certain operators are also very important.
  
The analytic functions (cf. [[Analytic function|Analytic function]]), i.e. functions locally representable by power series, form an important class, containing the rational functions. The analytic functions can be subdivided into the algebraic functions (cf. [[Algebraic function|Algebraic function]]), that is, functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940291.png" /> that can be given by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940292.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940293.png" /> is an irreducible polynomial with complex coefficients, and transcendental functions, that is, those which are not algebraic. With the concept of a [[Derivative|derivative]] one can associate the classes of functions that are differentiable a definite number of times, with the concept of an [[Integral|integral]] one can associate the classes of functions that are integrable in some sense or other, and with the concept of continuity one can associate the class of continuous functions. The [[Baire classes|Baire classes]] of functions are obtained by taking successive pointwise limits from the class of continuous functions (see also [[Borel function|Borel function]]). The definition of a [[Measurable function|measurable function]] is based on the concepts of a measurable set and a measure. The branch of the theory of functions that studies properties of functions associated with the concept of a measure is usually called the [[Metric theory of functions|metric theory of functions]].
+
The analytic functions (cf. [[Analytic function|Analytic function]]), i.e. functions locally representable by power series, form an important class, containing the rational functions. The analytic functions can be subdivided into the algebraic functions (cf. [[Algebraic function|Algebraic function]]), that is, functions $  y = f (x _{1} \dots x _{n} ) $
 +
that can be given by an equation $  P (x _{1} \dots x _{n} ,\  y) = 0 $,  
 +
where $  P $
 +
is an irreducible polynomial with complex coefficients, and transcendental functions, that is, those which are not algebraic. With the concept of a [[Derivative|derivative]] one can associate the classes of functions that are differentiable a definite number of times, with the concept of an [[Integral|integral]] one can associate the classes of functions that are integrable in some sense or other, and with the concept of continuity one can associate the class of continuous functions. The [[Baire classes|Baire classes]] of functions are obtained by taking successive pointwise limits from the class of continuous functions (see also [[Borel function|Borel function]]). The definition of a [[Measurable function|measurable function]] is based on the concepts of a measurable set and a measure. The branch of the theory of functions that studies properties of functions associated with the concept of a measure is usually called the [[Metric theory of functions|metric theory of functions]].
  
A function space arises as a collection of functions having certain general properties. Thus, all functions defined on the same set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940294.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940295.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940296.png" /> and, for example, Lebesgue measurable, continuous, or satisfying a [[Hölder condition|Hölder condition]] of given order, respectively, form vector spaces. Similarly, the spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940297.png" /> times (continuously-) differentiable functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940298.png" /> the infinitely-differentiable functions, the functions of compact support, the analytic functions, and many other classes of functions form vector spaces.
+
A function space arises as a collection of functions having certain general properties. Thus, all functions defined on the same set $  X $
 +
in the $  n $-
 +
dimensional Euclidean space $  \mathbf R ^{n} $
 +
and, for example, Lebesgue measurable, continuous, or satisfying a [[Hölder condition|Hölder condition]] of given order, respectively, form vector spaces. Similarly, the spaces of $  m $
 +
times (continuously-) differentiable functions, $  m = 1,\  2 \dots $
 +
the infinitely-differentiable functions, the functions of compact support, the analytic functions, and many other classes of functions form vector spaces.
  
In a number of vector spaces of functions one can introduce a norm. For example, in the space of continuous functions on a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940299.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940300.png" />, a norm is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940301.png" />; the normed space of continuous functions with this norm is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940302.png" />. Given the space of measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940303.png" />, defined on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940305.png" /> is a certain set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940306.png" /> is a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940307.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940309.png" /> is a measure defined on the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940310.png" />, by putting
+
In a number of vector spaces of functions one can introduce a norm. For example, in the space of continuous functions on a compact space $  X $,
 +
f: \  X \rightarrow \mathbf C $,  
 +
a norm is $  \| f \| = \mathop{\rm sup}\nolimits _{ {x \in X}} \  | f (x) | $;  
 +
the normed space of continuous functions with this norm is denoted by $  C (X) $.  
 +
Given the space of measurable functions $  f: \  X \rightarrow \mathbf C $,  
 +
defined on a space $  (X,\  \mathfrak S ,\  \mu ) $,  
 +
where $  X $
 +
is a certain set, $  \mathfrak S $
 +
is a certain $  \sigma $-
 +
algebra of subsets of $  X $
 +
and $  \mu $
 +
is a measure defined on the sets $  A \in \mathfrak S $,  
 +
by putting
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940311.png" /></td> </tr></table>
+
$$
 +
\| f \| _{p} \  = \  \left \{
  
one specifies a [[Norm|norm]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940312.png" /> on the set of functions for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940313.png" />. A function space with such a norm is usually called a Lebesgue function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940314.png" />. Among the other function spaces playing an important role in mathematical analysis one should mention a [[Hölder space|Hölder space]], a [[Nikol'skii space|Nikol'skii space]], an [[Orlicz space|Orlicz space]], and a [[Sobolev space|Sobolev space]]. All these spaces and certain generalizations of them are complete metric spaces, which to a large extent is important in studying many problems in the theory of functions itself as well as problems from related branches of mathematics. The relations between various norms for functions belonging simultaneously to various function spaces are studied in the theory of imbedding of function spaces (see [[Imbedding theorems|Imbedding theorems]]). An important property of the basic function spaces is that the set of infinitely-differentiable functions is dense in them, which enables one to study a number of properties of these function spaces on sufficiently-smooth functions, and to carry over the results to all functions in the space under consideration by taking limits.
+
\begin{array}{ll}
 +
\left ( \int\limits _{X} | f (x) | ^{p} \  d \mu \right ) ^{1/p}  &\  \textrm{ if } \  1 \leq p < + \infty ,  \\
 +
\mathop{\rm ess}\nolimits \  \mathop{\rm sup}\limits _ {x \in X} \  | f (x) |  &\  \textrm{ if } \  p = + \infty ,  \\
 +
\end{array}
 +
\right.
 +
$$
 +
 
 +
one specifies a [[Norm|norm]] $  \| f \| _{p} $
 +
on the set of functions for which $  \| f \| _{p} < + \infty $.  
 +
A function space with such a norm is usually called a Lebesgue function space $  L _{p} (X) $.  
 +
Among the other function spaces playing an important role in mathematical analysis one should mention a [[Hölder space|Hölder space]], a [[Nikol'skii space|Nikol'skii space]], an [[Orlicz space|Orlicz space]], and a [[Sobolev space|Sobolev space]]. All these spaces and certain generalizations of them are complete metric spaces, which to a large extent is important in studying many problems in the theory of functions itself as well as problems from related branches of mathematics. The relations between various norms for functions belonging simultaneously to various function spaces are studied in the theory of imbedding of function spaces (see [[Imbedding theorems|Imbedding theorems]]). An important property of the basic function spaces is that the set of infinitely-differentiable functions is dense in them, which enables one to study a number of properties of these function spaces on sufficiently-smooth functions, and to carry over the results to all functions in the space under consideration by taking limits.
  
 
==Dependence of functions.==
 
==Dependence of functions.==
This is a property of systems of functions generalizing the concept of their linear dependence and meaning that there are well-determined relations between the values of the functions in the given system; in particular, the values of one of them can be expressed in terms of the values of the others. For example, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940315.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940316.png" /> are dependent over the whole real line, since always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940317.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940318.png" /> be a [[Domain|domain]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940319.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940320.png" /> be its closure and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940321.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940322.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940323.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940324.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940325.png" />, are said to be dependent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940326.png" /> if there is a continuously-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940327.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940328.png" />, whose zeros form a [[Nowhere-dense set|nowhere-dense set]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940329.png" /> and such that the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940330.png" /> is identically zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940331.png" />.
+
This is a property of systems of functions generalizing the concept of their linear dependence and meaning that there are well-determined relations between the values of the functions in the given system; in particular, the values of one of them can be expressed in terms of the values of the others. For example, the functions $  f _{1} (x) = \sin ^{2} \  x $
 +
and  $  f _{2} (x) = \cos ^{2} \  x $
 +
are dependent over the whole real line, since always $  \sin ^{2} \  x = 1 - \cos ^{2} \  x $.  
 +
Let $  D $
 +
be a [[Domain|domain]] in $  \mathbf R ^{n} $,  
 +
let $  \overline{D}\; $
 +
be its closure and let $  f: \  \overline{D}\; \rightarrow \mathbf R ^{n} $,
 +
f (x) = \{ {y _{i} = f _{i} (x)} : {i = 1 \dots n} \} $,  
 +
$  x \in \overline{D}\; $.  
 +
The functions f _{i} $,  
 +
$  i = 1 \dots n $,  
 +
are said to be dependent in $  \overline{D}\; $
 +
if there is a continuously-differentiable function $  F (y) $,  
 +
$  y \in \mathbf R ^{n} $,  
 +
whose zeros form a [[Nowhere-dense set|nowhere-dense set]] in $  \mathbf R ^{n} $
 +
and such that the composite $  F \circ f $
 +
is identically zero on $  \overline{D}\; $.
 +
 
 +
Functions  $  f _{i} : \  G \rightarrow \mathbf R $,
 +
$  i = 1 \dots n $,
 +
are said to be dependent in the domain  $  G \subset \mathbf R ^{n} $
 +
if they are dependent in the closure  $  \overline{D}\; $
 +
of any domain  $  D $
 +
such that  $  \overline{D}\; \subset G $.
 +
 
 +
Functions  $  f _{i} $,
 +
$  i = 1 \dots n $,
 +
that are continuously differentiable in a domain  $  G \subset \mathbf R ^{n} $
 +
are dependent in  $  G $
 +
if and only if their [[Jacobian|Jacobian]]
  
Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940332.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940333.png" />, are said to be dependent in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940334.png" /> if they are dependent in the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940335.png" /> of any domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940336.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940337.png" />.
+
$$
  
Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940338.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940339.png" />, that are continuously differentiable in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940340.png" /> are dependent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940341.png" /> if and only if their [[Jacobian|Jacobian]]
+
\frac{\partial (f _{1} \dots f _{n} )}{\partial (x _{1} \dots x _{n} )}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940342.png" /></td> </tr></table>
+
$$
  
vanishes identically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940343.png" />.
+
vanishes identically on $  G $.
  
 
Now let
 
Now let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940344.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1}
 +
f _{i} : \  G \  \rightarrow \  \mathbf R ^{m} ,\ \
 +
i = 1 \dots m \leq n,\ \
 +
G \subset \mathbf R ^{n} .
 +
$$
  
If for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940345.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940346.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940347.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940348.png" />, there is an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940349.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940350.png" /> and a continuously-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940351.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940352.png" /> such that at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940353.png" />,
+
If for $  y _{i} = f _{i} (x) $,  
 +
$  x \in G $,  
 +
$  i = 1 \dots m $,  
 +
$  G \subset \mathbf R ^{n} $,  
 +
there is an open set $  \Gamma $
 +
in $  \mathbf R _{ {y _{1} \dots y _ {m - 1}}} ^ {m - 1} $
 +
and a continuously-differentiable function $  \Phi (y _{1} \dots y _{ {m - 1}} ) $
 +
on $  \Gamma $
 +
such that at any point $  x \in G $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940354.png" /></td> </tr></table>
+
$$
 +
(f _{1} (x) \dots f _{ {m - 1}} (x)) \  \in \  \Gamma
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940355.png" /></td> </tr></table>
+
$$
 +
\Phi (f _{1} (x) \dots f _{ {m - 1}} (x)) \  = f _{m} (x)
 +
$$
  
are satisfied, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940356.png" /> is said to be dependent in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940357.png" /> on the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940358.png" />.
+
are satisfied, then f _{m} $
 +
is said to be dependent in the set $  G $
 +
on the functions $  f _{1} \dots f _{ {m - 1}} $.
  
If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940359.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940360.png" />, are continuous in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940361.png" /> and if in a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940362.png" /> one of them depends on the others, then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940363.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940364.png" />, are dependent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940365.png" />.
+
If the functions f _{i} $,  
 +
$  i = 1 \dots m \leq n $,  
 +
are continuous in a domain $  G $
 +
and if in a neighbourhood of each point $  x \in G $
 +
one of them depends on the others, then the functions f _{i} $,  
 +
$  i = 1 \dots n $,  
 +
are dependent in $  G $.
  
If in a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940366.png" /> one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940367.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940368.png" />, which are continuously-differentiable in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940369.png" />, depends on the others, then at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940370.png" /> the rank of the Jacobi matrix
+
If in a neighbourhood of each point $  x \in G $
 +
one of the functions f _{i} $,  
 +
$  i = 1 \dots m \leq n $,  
 +
which are continuously-differentiable in a domain $  G \subset \mathbf R ^{n} $,  
 +
depends on the others, then at any point of $  G $
 +
the rank of the Jacobi matrix
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940371.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2}
 +
\left \|
  
is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940372.png" />, that is, the gradients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940373.png" /> are linearly dependent at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940374.png" />.
+
\frac{\partial f _ i}{\partial x _ j}
 +
\
 +
\right \| ,\ \
 +
i = 1 \dots m,\ \
 +
j = 1 \dots n,
 +
$$
  
Let the functions (1) be continuously differentiable in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940375.png" /> and let the rank of their Jacobi matrix (2) not exceed a certain number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940376.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940377.png" />, at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940378.png" />; suppose, moreover, that at a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940379.png" /> it is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940380.png" />. In other words, there are variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940381.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940382.png" /> such that
+
is less than  $  m $,  
 +
that is, the gradients  $  \nabla f _{1} \dots \nabla f _{m} $
 +
are linearly dependent at each point  $  x \in G $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940383.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
Let the functions (1) be continuously differentiable in a domain  $  G \subset \mathbf R ^{n} $
 +
and let the rank of their Jacobi matrix (2) not exceed a certain number  $  r $,
 +
$  1 \leq r < m \leq n $,
 +
at every point  $  x \in G $;  
 +
suppose, moreover, that at a certain point  $  x ^{(0)} \in G $
 +
it is equal to  $  r $.
 +
In other words, there are variables  $  x _{ {j _ 1}} \dots x _{ {j _ r}} $
 +
and functions  $  y _{ {i _ 1}} = f _{ {i _ 1}} (x) \dots y _{ {i _ r}} = f _{ {i _ r}} (x) $
 +
such that
  
Then there is no neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940384.png" /> in which any of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940385.png" /> depends on the others and there is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940386.png" /> such that each of the remaining functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940387.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940388.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940389.png" />, depends in this neighbourhood on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940390.png" />. In particular, if the gradients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940391.png" /> are linearly dependent at all points of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940392.png" /> and if at a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940393.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940394.png" /> of them that are linearly independent, and consequently one of them, for example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940395.png" />, is a linear combination of the others, then there is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940396.png" /> such that in this neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940397.png" /> depends on the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940398.png" />.
+
$$ \tag{3}
 +
\left .
 +
\frac{\partial (f _{ {i _ 1}} \dots
 +
f _{ {i _ r}} )}{\partial (x _{ {j _ 1}} \dots
 +
x _{ {j _ r}} )}
 +
 
 +
\right | _{ {x = x ^ (0)}} \  \neq \  0.
 +
$$
 +
 
 +
Then there is no neighbourhood of $  x ^{(0)} $
 +
in which any of the functions $  f _{ {i _ 1}} \dots f _{ {i _ r}} $
 +
depends on the others and there is a neighbourhood of $  x ^{(0)} $
 +
such that each of the remaining functions f _{i} $,  
 +
$  i \neq i _{k} $,  
 +
$  k = 1 \dots r $,  
 +
depends in this neighbourhood on $  f _{ {i _ 1}} \dots f _{ {i _ r}} $.  
 +
In particular, if the gradients $  \nabla f _{1} \dots \nabla f _{m} $
 +
are linearly dependent at all points of the domain $  G $
 +
and if at a certain point $  x ^{(0)} \in G $
 +
there are $  m - 1 $
 +
of them that are linearly independent, and consequently one of them, for example $  \nabla f _{m} $,  
 +
is a linear combination of the others, then there is a neighbourhood of $  x ^{(0)} $
 +
such that in this neighbourhood f _{m} $
 +
depends on the functions $  f _{1} \dots f _{ {m - 1}} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  "Einleitung in die Analysis des Unendlichen" , '''1''' , Springer  (1983)  (Translated from Latin)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Lobachevskii,  "Complete works" , '''5''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Dedekind,  "The nature and meaning of numbers" , Open Court  (1901)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Tarski,  "Introduction to logic and to the methodology of deductive sciences" , Oxford Univ. Press  (1946)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Frege,  "Funktion und Begriff" , H. Pohle  (1891)  ((Reprint: Kleine Schriften, G. Olms, 1967))</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Church,  "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.P. Yushkevich,  "The concept of function up to the middle of the nineteenth century"  ''Arch. History of Exact Sci.'' , '''16'''  (1977)  pp. 37–85  ''Istor.-Mat. Issled.'' , '''17'''  (1966)  pp. 123–150</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  F.A. Medvedev,  "Outlines of the history of the theory of functions of a real variable" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  "Einleitung in die Analysis des Unendlichen" , '''1''' , Springer  (1983)  (Translated from Latin)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Lobachevskii,  "Complete works" , '''5''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Dedekind,  "The nature and meaning of numbers" , Open Court  (1901)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Tarski,  "Introduction to logic and to the methodology of deductive sciences" , Oxford Univ. Press  (1946)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Frege,  "Funktion und Begriff" , H. Pohle  (1891)  ((Reprint: Kleine Schriften, G. Olms, 1967))</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Church,  "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.P. Yushkevich,  "The concept of function up to the middle of the nineteenth century"  ''Arch. History of Exact Sci.'' , '''16'''  (1977)  pp. 37–85  ''Istor.-Mat. Issled.'' , '''17'''  (1966)  pp. 123–150</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  F.A. Medvedev,  "Outlines of the history of the theory of functions of a real variable" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A pre-image is also called an inverse image. The restriction of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940399.png" /> to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940400.png" /> is generally denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940401.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940402.png" />; in axiomatic set theory, the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940403.png" /> is always written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940404.png" /> in order to avoid any misunderstanding. Strictly speaking, the definition of a many-valued function in the article doesn't make sense: in fact, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940405.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940406.png" /> (the set of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940407.png" />) is called a many-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041940/f041940408.png" />. Finally, the phrase [[Metric theory of functions|metric theory of functions]] is not often used in the West; there is no generally accepted name for this area of research.
+
A pre-image is also called an inverse image. The restriction of a function f $
 +
to a set $  A $
 +
is generally denoted by f \mid  _{A} $
 +
or f \mid  A $;  
 +
in axiomatic set theory, the image f (A) $
 +
is always written f {\  \prime\prime} A $
 +
in order to avoid any misunderstanding. Strictly speaking, the definition of a many-valued function in the article doesn't make sense: in fact, a function $  f : \  X \rightarrow Y $
 +
where $  Y = {\mathcal P} (Z) $(
 +
the set of subsets of $  Z $)  
 +
is called a many-valued function $  f : \  X \rightarrow Z $.  
 +
Finally, the phrase [[Metric theory of functions|metric theory of functions]] is not often used in the West; there is no generally accepted name for this area of research.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Mathematics, form and function" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan  (1968)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.C.M. van Rooy,  W.H. Schikhof,  "A second course on real functions" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.R. Gelbaum,  J.M.H. Olmsted,  "Counterexamples in analysis" , Holden-Day  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Mathematics, form and function" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan  (1968)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.C.M. van Rooy,  W.H. Schikhof,  "A second course on real functions" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.R. Gelbaum,  J.M.H. Olmsted,  "Counterexamples in analysis" , Holden-Day  (1964)</TD></TR></table>

Latest revision as of 22:06, 28 January 2020


One of the basic concepts in mathematics. Let two sets $X$ and $Y$ be given and suppose that to each element $x\in X$ corresponds an element $y\in Y$, which is denoted by $f(x)$. In this case one says that a function $f$ is given on $X$ (and also that the variable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$.

In ancient mathematics the idea of functional dependence was not expressed explicitly and was not an independent object of research, although a wide range of specific functional relations were known and were studied systematically. The concept of a function appears in a rudimentary form in the works of scholars in the Middle Ages, but only in the work of mathematicians in the 17th century, and primarily in those of P. Fermat, R. Descartes, I. Newton, and G. Leibniz, did it begin to take shape as an independent concept. The term "function" first appeared in works of Leibniz. Geometric, analytic and kinematic ideas were used to specify a function, but gradually the notion of a function as a certain analytic expression began to prevail. This was formulated in the 18th century in a precise form; J. Bernoulli's definition is that "a function of a variable quantity … is a number composed by some arbitrary method from the variable quantity and from constants" . L. Euler, having accepted this definition, wrote in his textbook on analysis that "all analysis of infinitesimals revolves around the variable quantities and their functions" [1]. Euler already had a more general approach to the concept of a function as dependence of one variable quantity on another. This point of view was developed further in the work of J. Fourier, N.I. Lobachevskii, P. Dirichlet, B. Bolzano, and A.L. Cauchy, where the notion of a function as a correspondence between two sets of numbers began to crystallize. So by 1834, Lobachevskii [2] was writing: "The general concept of a function requires that a function of x is a number which is given for each x and gradually changes with x. The value of a function can be given either by an analytic expression or by a condition which gives a means of testing all numbers and choosing one of them; or finally a dependence can exist and remain unknown" . The definition of a function as a correspondence between two arbitrary sets (not necessarily consisting of numbers) was formulated by R. Dedekind in 1887 [3].

The concept of a correspondence, and consequently also the concept of a function, sometimes leads to other concepts (to a set [4], a relation [5] or some other set-theoretical or mathematico-logical concepts [6]) and is sometimes taken as a primary, undefined, concept [7]. A. Church, for example, expressed the view that: "In the end it is necessary to consider the concept of a function — or some similar concept, for example the concept of a class — as primitive or undefinable" [8]. (For more information see [9], [10].)

The concept of a function considered below is based on the concept of a set and of the simplest operations on sets.

One says that the number of elements of a set $ A $ is equal to 1 or that the set $ A $ consists of one element if it contains an element $ a $ and no others (in other words, if after deleting the set $ \{ a \} $ from $ A $ one obtains the empty set). A non-empty set $ A $ is called a set with two elements, or a pair, $ A = \{ a,\ b \} $, if after deleting a set consisting of only one element $ a \in A $ there remains a set also consisting of one element $ b \in A $( this definition does not depend on the choice of the chosen element $ a \in A $).

If a pair $ A = \{ a,\ b \} $ is given, then the pair $ \{ a,\ \{ a,\ b \} \} $ is called the ordered pair of elements $ a \in A $ and $ b \in A $ and is denoted by $ (a,\ b) $. The element $ a \in A $ is called its first element and $ b \in A $ is called the second element.

Given sets $ X $ and $ Y $, the set of all ordered pairs $ (x,\ y) $, $ x \in X $, $ y \in Y $, is called the product of the sets $ X $ and $ Y $ and is denoted by $ X \times Y $. It is not assumed that $ X $ is different from $ Y $, that is, it is possible that $ X = Y $.

Each set $ f = \{ (x,\ y) \} $ of ordered pairs $ (x,\ y) $, $ x \in X $, $ y \in Y $, such that, if $ (x ^ \prime ,\ y ^ \prime ) \in f $ and $ (x ^{\prime\prime} ,\ y ^{\prime\prime} ) \in f $, then $ y ^ \prime \neq y ^{\prime\prime} $ implies that $ x ^ \prime \neq x ^{\prime\prime} $, is called a function or, what is the same thing, a mapping.

As well as the terms "function" and "mapping" one uses in certain situations the terms transformation , morphism , correspondence , which are equivalent to them.

The set of all first elements of ordered pairs $ (x,\ y) $ of a given function $ f $ is called the domain of definition (or the set of definition) of this function and is denoted by $ X _{f} $, and the set of all second elements is called the range of values (the set of values) and is denoted by $ Y _{f} $. The set of ordered pairs itself, $ f = \{ (x,\ y) \} $, considered as a subset of the product $ X \times Y $, is called the graph of $ f $.

The element $ x \in X $ is called the argument of the function, or the independent variable, and the element $ y \in Y $ is called the dependent variable.

If $ f = \{ (x,\ y) \} $ is a function, then one writes $ f: \ X _{f} \rightarrow Y $ and says that $ f $ maps the set $ X _{f} $ into the set $ Y $. In the case $ X = X _{f} $ one simply writes $ f: \ X \rightarrow Y $.

If $ f: \ X \rightarrow Y $ is a function and $ (x,\ y) \in f $, then one writes $ y = f (x) $( sometimes simply $ y = fx $ or $ y = x f \ $) and also $ f: \ x \mapsto y $, $ x \in X $, $ y \in Y $, and says that the function $ f $ puts the element $ y $ in correspondence with the element $ x $( the mapping $ f $ maps $ x $ to $ y $) or, what is the same, the element $ y $ corresponds to the element $ x $. In this case one also says that $ y $ is the value of $ f $ at the point $ x $, or that $ y $ is the image of the element $ x $ under $ f $.

As well as the symbol $ f (x _{0} ) $ one also uses the notation $ f (x) \mid _{ {x = x _ 0}} $ for denoting the value of $ f $ at $ x _{0} $.

Sometimes the function $ f $ itself is denoted by the symbol $ f (x) $. Denoting both the function $ f: \ X \rightarrow Y $ and its value at the point $ x \in X $ by the same symbol $ f (x) $ does not usually lead to misunderstanding, since in any particular case, as a rule, it is always clear what one is talking about. The notation $ f (x) $ often turns out to be more convenient than the notation $ f: \ x \mapsto y $ in computations. For example, writing $ f (x) = x ^{2} $ is more convenient and simpler to use in analytic manipulations than writing $ f: \ x \mapsto x ^{2} $.

Given $ y \in Y $, the set of all elements $ x \in X $ such that $ f (x) = y $ is called the pre-image of the element $ y $ and is denoted by $ f ^ {\ -1} (y) $. Thus,

$$ f ^ {\ -1} (y) \ = \ \{ {x} : {x \in X,\ f (x) = y} \} . $$

Obviously, if $ y \in Y \setminus Y _{f} $, then $ f ^ {\ -1} (y) = \emptyset $, the empty set.

Let a mapping $ f: \ X \rightarrow Y $ be given. In other words, to each $ x \in X $ corresponds a unique element $ y \in Y $ and to each $ y \in Y _{f} \subset Y $ corresponds at least one element $ x \in X $. If $ Y = X $, one says that $ f $ maps the set $ X $ into itself. If $ Y = Y _{f} $, that is, if $ Y $ coincides with the range of $ f $, then one says that $ f $ maps $ X $ onto the set $ Y $ or that $ f $ is a surjective mapping, or, more concisely, it is a surjection. Thus, a mapping $ f: \ X \rightarrow Y $ is a surjection if for each element $ y \in Y $ there is at least one element $ x \in X $ such that $ f (x) = y $.

If under a mapping $ f: \ X \rightarrow Y $ different elements $ y \in Y $ correspond to different elements $ x \in X $, that is, if $ x ^ \prime \neq x ^{\prime\prime} $ implies $ f (x ^ \prime ) \neq f (x ^{\prime\prime} ) $, then $ f $ is said to be a one-to-one mapping of $ X $ into $ Y $ and also a univalent mapping or an injection. Thus, a mapping $ f: \ X \rightarrow Y $ is univalent (injective) if and only if the pre-image of each element $ y $ belonging to the range of $ f $, that is, $ y \in Y _{f} $, consists precisely of one element. If the mapping $ f: \ X \rightarrow Y $ is simultaneously one-to-one and onto the set $ Y $( see One-to-one correspondence), that is, is at the same time injective and surjective, then it is called a bijective mapping or a bijection.

If $ f: \ X \rightarrow Y $ and $ A \subset X $, then the set

$$ S \ = \ \{ {y} : {y \in Y,\ y = f (x),\ x \in A} \} , $$

that is, the set of all those $ y $ such that there is at least one element of the subset $ A $ of $ X $ which is mapped to $ y $ by $ f $, is called the image of the subset $ A $, and one writes $ S = f (A) $. In particular, always $ Y _{f} = f (X) $. The following relations are true for the images of sets $ A \subset X $ and $ B \subset X $:

$$ f (A \cup B) \ = \ f (A) \cup f (B), $$

$$ f (A \cap B) \ \subset \ f (A) \cap f (B), $$

$$ f (A) \setminus f (B) \ \subset \ f (A \setminus B), $$

and if $ A \subset B $, then $ f (A) \subset f (B) $.

If $ f: \ X \rightarrow Y $ and $ S \subset Y $, then the set

$$ A \ = \ \{ {x} : {x \in X,\ f (x) \in S} \} $$

is called the pre-image of the set $ S $ and one writes $ A = f ^ {\ -1} (S) $. Thus, the pre-image of a set $ S $ consists of all those elements $ x \in X $ which are mapped to elements of $ S $ under $ f $, or what is the same thing, it consists of all pre-images of elements $ y \in S $: $ f ^ {\ -1} (S) = \cup _{ {y \in S}} f ^ {\ -1} (y) $. For the pre-images of sets $ S \subset Y $ and $ T \subset Y $ the relations

$$ f ^ {\ -1} (S \cup T) \ = \ f ^ {\ -1} (S) \cup f ^ {\ -1} (T), $$

$$ f ^ {\ -1} (S \cap T) \ = \ f ^ {\ -1} (S) \cap f ^ {\ -1} (T), $$

$$ f ^ {\ -1} (S \setminus T) \ = \ f ^ {\ -1} (S) \setminus f ^ {\ -1} (T) $$

are true, and if $ S \subset T $, then $ f ^ {\ -1} (S) \subset f ^ {\ -1} (T) $.

If $ A \subset X $, then the function $ f: \ X \rightarrow Y $ generates in a natural way a function defined on $ A $ under which $ f (x) $ corresponds to the element $ x \in A $. This function is called the restriction of the function $ f $ to the set $ A $ and is sometimes denoted by $ f _{A} $. Thus $ f _{A} : \ A \rightarrow Y $ and for any $ x \in A $ one has $ f _{A} : \ x \mapsto f (x) $. If the set $ A $ does not coincide with the set $ X $, then the restriction $ f _{A} $ of $ f $ to $ A $ may have a different domain of definition than $ f $ and, consequently, is different from $ f $.

If $ f: \ X \rightarrow Y $, if each element $ y \in Y _{f} $ is a certain set of elements $ y = \{ z \} $, and if, moreover, among these sets there is at least one set consisting of more than one element, then $ f $ is called a multi-valued function (sometimes, many-valued function). Further, the elements of the set $ f (x) = \{ z \} $ are often called the values of $ f $ at $ x $. If each set $ f (x) $, $ x \in X $, consists of only one element, then the function $ f $ is also called a single-valued function.

If $ f: \ X \rightarrow Y $ and $ g: \ Y \rightarrow Z $, then the function $ F: \ X \rightarrow Z $ defined for each $ x \in X $ by the formula $ F (x) = g (f (x)) $ is called the composition (superposition) of the functions $ f $ and $ g $, also the composite function, and is denoted by $ g \circ f $.

Let a function $ f: \ X \rightarrow Y $ be given and let $ Y _{f} $ be its range. The set of all possible ordered pairs of the form $ (y,\ f ^ {\ -1} (y)) $, $ y \in Y _{f} $, forms a function, called the inverse function of $ f $ and denoted by $ f ^ {\ -1} $. Under the inverse function $ f ^ {\ -1} $, to each $ y \in Y _{f} $ corresponds the pre-image $ f ^ {\ -1} (y) $, that is, a certain set of elements. So the inverse function is, generally speaking, a multi-valued function. If a mapping $ f: \ X \rightarrow Y $ is injective, then the inverse mapping is a single-valued function and maps the range $ Y _{f} $ of $ f $ onto the domain of definition $ X $ of $ f $.

Functions on numbers.

An important class of functions is that of the complex-valued functions $ f: \ X \rightarrow Y $, $ Y \subset \mathbf C $, where $ \mathbf C $ is the set of all complex numbers. One can carry out various arithmetical operations on complex-valued functions. If two given complex-valued functions $ f $ and $ g $ are defined on the same set $ X $ and if $ \lambda $ is a complex number, then the function $ \lambda f $ is defined as the function taking the value $ \lambda f (x) $ at each point $ x \in X $; the function $ f + g $ is the function taking the value $ f (x) + g (x) $ at each point; the function $ fg $ is the function taking the value $ f (x) g (x) $ at each point; and, finally, $ f/g $ is the function equal to $ f (x)/g (x) $ at each point $ x \in X $( which, of course, makes sense only when $ g (x) \neq 0 $).

A function $ f: \ X \rightarrow \mathbf R $ is called a real-valued function ( $ \mathbf R $ is the set of real numbers). A real-valued function $ f: \ X \rightarrow \mathbf R $ is said to be bounded from above (bounded from below) on the set $ X $ if its range is bounded from above (bounded from below). In other words, a function $ f: \ X \rightarrow \mathbf R $ is bounded from above (bounded from below) on $ X $ if there is a constant $ c \in \mathbf R $ such that for every $ x \in X $ the inequality $ f (x) \leq c $ is satisfied (the inequality $ f (x) \geq c $ is satisfied, respectively). A function $ f $ that is both bounded from above and from below on $ X $ is simply said to be bounded on $ X $. An upper (lower) bound of the range of $ f: \ X \rightarrow \mathbf R $ is an upper (lower) bound of the function $ f $.

A major role in mathematical analysis is played by functions on numbers or, more precisely, complex-valued functions of a complex variable, that is, functions $ f: \ X \rightarrow Y $, where $ X,\ Y \subset \mathbf C $. If the domain of definition of such a function and its range are both subsets of the real numbers, then this function is called a real function, or, more precisely, a real-valued function of a real variable. The generalization of the concept of a function on numbers is, first of all, a complex-valued function of several complex variables, called a complex function of several variables. A further generalization of a function on numbers is a vector-valued function (see Vector function) and, in general, a function for which the domain of definition and the range are provided with definite structures. For example, if the ranges of functions belong to a certain vector space, then such functions can be added; if they belong to a ring, then the functions can be added and multiplied; if they belong to a set which is ordered in some specific way, then one can generalize to these functions the idea of boundedness, upper and lower bounds, etc. The presence of topological structures on the sets $ X $ and $ Y $ enables one to introduce the concept of a continuous function $ f: \ X \rightarrow Y $. In the case when $ X $ and $ Y $ are topological vector spaces one introduces the concept of differentiability for a function $ f: \ X \rightarrow Y $( cf. Differentiable function).

Methods for specifying functions.

Functions on numbers (and certain generalizations of them) can be given by formulas. This is the analytic method for specifying functions. For this one uses a certain supply of functions which have been studied and have a special notation (primarily the elementary functions), algebraic operations, composition of functions and limit transitions (which includes operations of mathematical analysis such as differentiation, integration, summing series), for example:

$$ y \ = \ ax + b,\ \ y \ = \ ax ^{2} ,\ \ y \ = \ 1 + \sqrt { \mathop{\rm log}\nolimits \ \cos \ 2 \pi x} , $$

$$ \zeta (z) \ = \ \sum _ {n = 1} ^ \infty { \frac{1}{n ^ z} } , $$

$$ P _{n} (x) \ = \ { \frac{1}{2 ^{n} n!} } \frac{d ^{n} (x ^{2} - 1) ^ n}{dx ^ n} , $$

$$ I ( \alpha ,\ \beta ) \ = \ \int\limits _{0} ^ {+ \infty} e ^ {- \alpha x} { \frac{\sin \ \beta x}{x} } \ dx. $$

The class of functions that are presented, in a well-determined sense, as the sum of series, even only as sums of trigonometric series, is very wide. A function can be given analytically either in an explicit form, that is, by a formula of the type $ y = f (x) $, or as an implicit function, that is, by an equation of the type $ F (x,\ y) = 0 $. Sometimes the function is given with the aid of several formulas, for example,

$$ \tag{*} f (x) \ = \ \left \{ \begin{array}{ll} 2 ^{x} \ &\textrm{ if } \ x > 0, \\ 0 \ &\textrm{ if } \ x = 0, \\ x - 1 \ &\textrm{ if } \ x < 0. \\ \end{array} \right .$$

A function can also be given by using a description of the correspondence. Let, for example, the number 1 correspond to every $ x > 0 $, the number 0 to the number 0, and the number $ -1 $ to every $ x < 0 $. As a result one obtains a function defined on the whole real line and taking three values: $ 1,\ 0,\ -1 $. This function has the special notation $ \mathop{\rm sign}\nolimits \ x $( or $ \mathop{\rm sgn}\nolimits \ x $). Another example: the number 1 corresponds to each rational number and the number 0 to each irrational number. The function obtained is called the Dirichlet function. The same function can be given in different ways; for example, the function $ \mathop{\rm sign}\nolimits \ x $ and the Dirichlet function can be defined not only by verbal descriptions but also by formulas.

Every formula is a symbolic notation of a certain previously-described correspondence, so that in the end there is no fundamental difference between specifying a function by a formula or by a verbal description of the correspondence; this difference is superficial. It should be borne in mind that every function newly defined by some means or other can, if a special notation is introduced for it, serve to define other functions by using formulas including this new symbol. However, for an analytic representation of a function the supply of functions and operations that are to be used in the formulas is very essential; usually one tries to make this supply as small as possible and chooses the functions and operations as simply as possible in a certain well-determined sense.

When one is concerned with real-valued functions of a single real variable, then to give an intuitive representation of the nature of the functional dependence one often constructs the graph of the function in the coordinate plane, in other words, given a function $ f: \ X \rightarrow \mathbf R $, $ X \subset \mathbf R $, one considers the set of points $ (x,\ f (x)) $, $ x \in X $, in the $ (x,\ y) $- plane. $ y = x - 1 $ $ y = 2 ^{x} $

Figure: f041940a

Figure: f041940b

Thus, the graph of the function (*) has the form depicted in Fig. a, the graph of the function $ \mathop{\rm sign}\nolimits \ x $ is Fig. b, and the graph of the function $ y = 1 + \sqrt { \mathop{\rm log}\nolimits \ \cos \ 2 \pi x} $ consists of the isolated points in Fig. c.

Figure: f041940c

The representation of a function by a graph can also serve to reveal a functional dependence. This revelation is approximate because, in practice, the measurement of the intervals can be carried out only with a definite degree of accuracy, without mentioning the fact that, when the domain of definition of the function is unbounded, it is in principle impossible to draw it on the coordinate plane.

A tabular method is also extensively used for representing functions on numbers, either in the form of prepared tables of values of the function at definite points, or by introducing this data into a machine memory, or by making a program to calculate the values on a computer.

The classification of complex or real functions.

The simplest complex functions are the elementary functions, among which one distinguishes the algebraic polynomials, the trigonometric polynomials and also the rational functions (cf. Rational function). The special role of these functions is that one of the methods for studying and using more general functions is based on approximating them by algebraic polynomials, by trigonometric polynomials or by rational functions, as well as by functions composed from these functions in some well-determined way (see Spline). The branch of the theory of functions that studies the approximation of functions by collections of functions that are simple in a certain sense is called approximation theory. In this theory approximations of functions by linear aggregates of eigen functions of certain operators are also very important.

The analytic functions (cf. Analytic function), i.e. functions locally representable by power series, form an important class, containing the rational functions. The analytic functions can be subdivided into the algebraic functions (cf. Algebraic function), that is, functions $ y = f (x _{1} \dots x _{n} ) $ that can be given by an equation $ P (x _{1} \dots x _{n} ,\ y) = 0 $, where $ P $ is an irreducible polynomial with complex coefficients, and transcendental functions, that is, those which are not algebraic. With the concept of a derivative one can associate the classes of functions that are differentiable a definite number of times, with the concept of an integral one can associate the classes of functions that are integrable in some sense or other, and with the concept of continuity one can associate the class of continuous functions. The Baire classes of functions are obtained by taking successive pointwise limits from the class of continuous functions (see also Borel function). The definition of a measurable function is based on the concepts of a measurable set and a measure. The branch of the theory of functions that studies properties of functions associated with the concept of a measure is usually called the metric theory of functions.

A function space arises as a collection of functions having certain general properties. Thus, all functions defined on the same set $ X $ in the $ n $- dimensional Euclidean space $ \mathbf R ^{n} $ and, for example, Lebesgue measurable, continuous, or satisfying a Hölder condition of given order, respectively, form vector spaces. Similarly, the spaces of $ m $ times (continuously-) differentiable functions, $ m = 1,\ 2 \dots $ the infinitely-differentiable functions, the functions of compact support, the analytic functions, and many other classes of functions form vector spaces.

In a number of vector spaces of functions one can introduce a norm. For example, in the space of continuous functions on a compact space $ X $, $ f: \ X \rightarrow \mathbf C $, a norm is $ \| f \| = \mathop{\rm sup}\nolimits _{ {x \in X}} \ | f (x) | $; the normed space of continuous functions with this norm is denoted by $ C (X) $. Given the space of measurable functions $ f: \ X \rightarrow \mathbf C $, defined on a space $ (X,\ \mathfrak S ,\ \mu ) $, where $ X $ is a certain set, $ \mathfrak S $ is a certain $ \sigma $- algebra of subsets of $ X $ and $ \mu $ is a measure defined on the sets $ A \in \mathfrak S $, by putting

$$ \| f \| _{p} \ = \ \left \{ \begin{array}{ll} \left ( \int\limits _{X} | f (x) | ^{p} \ d \mu \right ) ^{1/p} &\ \textrm{ if } \ 1 \leq p < + \infty , \\ \mathop{\rm ess}\nolimits \ \mathop{\rm sup}\limits _ {x \in X} \ | f (x) | &\ \textrm{ if } \ p = + \infty , \\ \end{array} \right. $$

one specifies a norm $ \| f \| _{p} $ on the set of functions for which $ \| f \| _{p} < + \infty $. A function space with such a norm is usually called a Lebesgue function space $ L _{p} (X) $. Among the other function spaces playing an important role in mathematical analysis one should mention a Hölder space, a Nikol'skii space, an Orlicz space, and a Sobolev space. All these spaces and certain generalizations of them are complete metric spaces, which to a large extent is important in studying many problems in the theory of functions itself as well as problems from related branches of mathematics. The relations between various norms for functions belonging simultaneously to various function spaces are studied in the theory of imbedding of function spaces (see Imbedding theorems). An important property of the basic function spaces is that the set of infinitely-differentiable functions is dense in them, which enables one to study a number of properties of these function spaces on sufficiently-smooth functions, and to carry over the results to all functions in the space under consideration by taking limits.

Dependence of functions.

This is a property of systems of functions generalizing the concept of their linear dependence and meaning that there are well-determined relations between the values of the functions in the given system; in particular, the values of one of them can be expressed in terms of the values of the others. For example, the functions $ f _{1} (x) = \sin ^{2} \ x $ and $ f _{2} (x) = \cos ^{2} \ x $ are dependent over the whole real line, since always $ \sin ^{2} \ x = 1 - \cos ^{2} \ x $. Let $ D $ be a domain in $ \mathbf R ^{n} $, let $ \overline{D}\; $ be its closure and let $ f: \ \overline{D}\; \rightarrow \mathbf R ^{n} $, $ f (x) = \{ {y _{i} = f _{i} (x)} : {i = 1 \dots n} \} $, $ x \in \overline{D}\; $. The functions $ f _{i} $, $ i = 1 \dots n $, are said to be dependent in $ \overline{D}\; $ if there is a continuously-differentiable function $ F (y) $, $ y \in \mathbf R ^{n} $, whose zeros form a nowhere-dense set in $ \mathbf R ^{n} $ and such that the composite $ F \circ f $ is identically zero on $ \overline{D}\; $.

Functions $ f _{i} : \ G \rightarrow \mathbf R $, $ i = 1 \dots n $, are said to be dependent in the domain $ G \subset \mathbf R ^{n} $ if they are dependent in the closure $ \overline{D}\; $ of any domain $ D $ such that $ \overline{D}\; \subset G $.

Functions $ f _{i} $, $ i = 1 \dots n $, that are continuously differentiable in a domain $ G \subset \mathbf R ^{n} $ are dependent in $ G $ if and only if their Jacobian

$$ \frac{\partial (f _{1} \dots f _{n} )}{\partial (x _{1} \dots x _{n} )} $$

vanishes identically on $ G $.

Now let

$$ \tag{1} f _{i} : \ G \ \rightarrow \ \mathbf R ^{m} ,\ \ i = 1 \dots m \leq n,\ \ G \subset \mathbf R ^{n} . $$

If for $ y _{i} = f _{i} (x) $, $ x \in G $, $ i = 1 \dots m $, $ G \subset \mathbf R ^{n} $, there is an open set $ \Gamma $ in $ \mathbf R _{ {y _{1} \dots y _ {m - 1}}} ^ {m - 1} $ and a continuously-differentiable function $ \Phi (y _{1} \dots y _{ {m - 1}} ) $ on $ \Gamma $ such that at any point $ x \in G $,

$$ (f _{1} (x) \dots f _{ {m - 1}} (x)) \ \in \ \Gamma $$

and

$$ \Phi (f _{1} (x) \dots f _{ {m - 1}} (x)) \ = \ f _{m} (x) $$

are satisfied, then $ f _{m} $ is said to be dependent in the set $ G $ on the functions $ f _{1} \dots f _{ {m - 1}} $.

If the functions $ f _{i} $, $ i = 1 \dots m \leq n $, are continuous in a domain $ G $ and if in a neighbourhood of each point $ x \in G $ one of them depends on the others, then the functions $ f _{i} $, $ i = 1 \dots n $, are dependent in $ G $.

If in a neighbourhood of each point $ x \in G $ one of the functions $ f _{i} $, $ i = 1 \dots m \leq n $, which are continuously-differentiable in a domain $ G \subset \mathbf R ^{n} $, depends on the others, then at any point of $ G $ the rank of the Jacobi matrix

$$ \tag{2} \left \| \frac{\partial f _ i}{\partial x _ j} \ \right \| ,\ \ i = 1 \dots m,\ \ j = 1 \dots n, $$

is less than $ m $, that is, the gradients $ \nabla f _{1} \dots \nabla f _{m} $ are linearly dependent at each point $ x \in G $.

Let the functions (1) be continuously differentiable in a domain $ G \subset \mathbf R ^{n} $ and let the rank of their Jacobi matrix (2) not exceed a certain number $ r $, $ 1 \leq r < m \leq n $, at every point $ x \in G $; suppose, moreover, that at a certain point $ x ^{(0)} \in G $ it is equal to $ r $. In other words, there are variables $ x _{ {j _ 1}} \dots x _{ {j _ r}} $ and functions $ y _{ {i _ 1}} = f _{ {i _ 1}} (x) \dots y _{ {i _ r}} = f _{ {i _ r}} (x) $ such that

$$ \tag{3} \left . \frac{\partial (f _{ {i _ 1}} \dots f _{ {i _ r}} )}{\partial (x _{ {j _ 1}} \dots x _{ {j _ r}} )} \right | _{ {x = x ^ (0)}} \ \neq \ 0. $$

Then there is no neighbourhood of $ x ^{(0)} $ in which any of the functions $ f _{ {i _ 1}} \dots f _{ {i _ r}} $ depends on the others and there is a neighbourhood of $ x ^{(0)} $ such that each of the remaining functions $ f _{i} $, $ i \neq i _{k} $, $ k = 1 \dots r $, depends in this neighbourhood on $ f _{ {i _ 1}} \dots f _{ {i _ r}} $. In particular, if the gradients $ \nabla f _{1} \dots \nabla f _{m} $ are linearly dependent at all points of the domain $ G $ and if at a certain point $ x ^{(0)} \in G $ there are $ m - 1 $ of them that are linearly independent, and consequently one of them, for example $ \nabla f _{m} $, is a linear combination of the others, then there is a neighbourhood of $ x ^{(0)} $ such that in this neighbourhood $ f _{m} $ depends on the functions $ f _{1} \dots f _{ {m - 1}} $.

References

[1] L. Euler, "Einleitung in die Analysis des Unendlichen" , 1 , Springer (1983) (Translated from Latin)
[2] N.I. Lobachevskii, "Complete works" , 5 , Moscow-Leningrad (1951) (In Russian)
[3] R. Dedekind, "The nature and meaning of numbers" , Open Court (1901) (Translated from German)
[4] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[5] A. Tarski, "Introduction to logic and to the methodology of deductive sciences" , Oxford Univ. Press (1946) (Translated from German)
[6] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
[7] G. Frege, "Funktion und Begriff" , H. Pohle (1891) ((Reprint: Kleine Schriften, G. Olms, 1967))
[8] A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)
[9] A.P. Yushkevich, "The concept of function up to the middle of the nineteenth century" Arch. History of Exact Sci. , 16 (1977) pp. 37–85 Istor.-Mat. Issled. , 17 (1966) pp. 123–150
[10] F.A. Medvedev, "Outlines of the history of the theory of functions of a real variable" , Moscow (1975) (In Russian)

Comments

A pre-image is also called an inverse image. The restriction of a function $ f $ to a set $ A $ is generally denoted by $ f \mid _{A} $ or $ f \mid A $; in axiomatic set theory, the image $ f (A) $ is always written $ f {\ \prime\prime} A $ in order to avoid any misunderstanding. Strictly speaking, the definition of a many-valued function in the article doesn't make sense: in fact, a function $ f : \ X \rightarrow Y $ where $ Y = {\mathcal P} (Z) $( the set of subsets of $ Z $) is called a many-valued function $ f : \ X \rightarrow Z $. Finally, the phrase metric theory of functions is not often used in the West; there is no generally accepted name for this area of research.

References

[a1] S. MacLane, "Mathematics, form and function" , Springer (1986)
[a2] H.L. Royden, "Real analysis", Macmillan (1968) pp. Chapt. 5
[a3] A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982)
[a4] B.R. Gelbaum, J.M.H. Olmsted, "Counterexamples in analysis" , Holden-Day (1964)
How to Cite This Entry:
Function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function&oldid=44361
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article