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A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i0505001.png" /> defined on a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i0505002.png" /> of real numbers such that the condition
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<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/i/i050/i050500/i0505003.png" /></td> </tr></table>
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implies
+
A real-valued function  $  f $
 +
defined on a certain set  $  E $
 +
of real numbers such that the condition
  
<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/i/i050/i050500/i0505004.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  < x  ^ {\prime\prime} ,\ \
 +
x  ^  \prime  , x  ^ {\prime\prime}  \in  E
 +
$$
  
Such functions are sometimes called strictly increasing functions, the term  "increasing functions"  being reserved for functions which, for such given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i0505005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i0505006.png" />, merely satisfy the condition
+
implies
  
<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/i/i050/i050500/i0505007.png" /></td> </tr></table>
+
$$
 +
f ( x  ^  \prime  )  < f ( x  ^ {\prime\prime} ).
 +
$$
  
(non-decreasing functions). The inverse function of any strictly increasing function is single-valued and is also strictly increasing. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i0505008.png" /> is a right-sided (or left-sided) limit point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i0505009.png" /> (cf. [[Limit point of a set|Limit point of a set]]), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050010.png" /> is a non-decreasing function and if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050011.png" /> is bounded from below — or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050012.png" /> is bounded from above — then, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050013.png" /> (or, correspondingly, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050014.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050015.png" />, the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050016.png" /> will have a finite limit; if the set is not bounded from below (or, correspondingly, from above), the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050017.png" /> have an infinite limit equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050018.png" /> (or, correspondingly, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050019.png" />).
+
Such functions are sometimes called strictly increasing functions, the term  "increasing functions" being reserved for functions which, for such given  $  x  ^  \prime  $
 +
and $  x  ^ {\prime\prime} $,  
 +
merely satisfy the condition
  
 +
$$
 +
f ( x  ^  \prime  )  \leq  f ( x  ^ {\prime\prime} )
 +
$$
  
 +
(non-decreasing functions). The inverse function of any strictly increasing function is single-valued and is also strictly increasing. If  $  x _ {0} $
 +
is a right-sided (or left-sided) limit point of the set  $  E $(
 +
cf. [[Limit point of a set|Limit point of a set]]), if  $  f $
 +
is a non-decreasing function and if the set  $  A = \{ {y } : {y = f( x),  x > x _ {0} ,  x \in E } \} $
 +
is bounded from below — or if  $  \{ {y } : {y = f( x),  x < x _ {0} ,  x \in E } \} $
 +
is bounded from above — then, as  $  x \rightarrow x _ {0} + $(
 +
or, correspondingly, as  $  x \rightarrow x _ {0} - $),
 +
$  x \in E $,
 +
the values  $  f( x) $
 +
will have a finite limit; if the set is not bounded from below (or, correspondingly, from above), the values  $  f ( x) $
 +
have an infinite limit equal to  $  - \infty $(
 +
or, correspondingly, to  $  + \infty $).
  
 
====Comments====
 
====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050020.png" /> is non-decreasing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050022.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050023.png" /> referred to above is automatically bounded from below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050024.png" />, unless it is empty. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050025.png" /> is a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050026.png" />, then the right-hand limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050027.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050028.png" /> is simply the infimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050500/i05050029.png" />:
+
If $  f $
 +
is non-decreasing on $  E $
 +
and $  x _ {0} \in E $,  
 +
then the set $  A $
 +
referred to above is automatically bounded from below by $  f ( x _ {0} ) $,  
 +
unless it is empty. If, in addition, $  x _ {0} $
 +
is a limit point of $  \{ {x \in E } : {x > x _ {0} } \} $,  
 +
then the right-hand limit of $  f $
 +
at $  x _ {0} $
 +
is simply the infimum of $  A $:
  
<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/i/i050/i050500/i05050030.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \downarrow x _ {0} }  f ( x)  = \inf  A .
 +
$$
  
 
Similar remarks hold for left-hand limits.
 
Similar remarks hold for left-hand limits.

Latest revision as of 22:12, 5 June 2020


A real-valued function $ f $ defined on a certain set $ E $ of real numbers such that the condition

$$ x ^ \prime < x ^ {\prime\prime} ,\ \ x ^ \prime , x ^ {\prime\prime} \in E $$

implies

$$ f ( x ^ \prime ) < f ( x ^ {\prime\prime} ). $$

Such functions are sometimes called strictly increasing functions, the term "increasing functions" being reserved for functions which, for such given $ x ^ \prime $ and $ x ^ {\prime\prime} $, merely satisfy the condition

$$ f ( x ^ \prime ) \leq f ( x ^ {\prime\prime} ) $$

(non-decreasing functions). The inverse function of any strictly increasing function is single-valued and is also strictly increasing. If $ x _ {0} $ is a right-sided (or left-sided) limit point of the set $ E $( cf. Limit point of a set), if $ f $ is a non-decreasing function and if the set $ A = \{ {y } : {y = f( x), x > x _ {0} , x \in E } \} $ is bounded from below — or if $ \{ {y } : {y = f( x), x < x _ {0} , x \in E } \} $ is bounded from above — then, as $ x \rightarrow x _ {0} + $( or, correspondingly, as $ x \rightarrow x _ {0} - $), $ x \in E $, the values $ f( x) $ will have a finite limit; if the set is not bounded from below (or, correspondingly, from above), the values $ f ( x) $ have an infinite limit equal to $ - \infty $( or, correspondingly, to $ + \infty $).

Comments

If $ f $ is non-decreasing on $ E $ and $ x _ {0} \in E $, then the set $ A $ referred to above is automatically bounded from below by $ f ( x _ {0} ) $, unless it is empty. If, in addition, $ x _ {0} $ is a limit point of $ \{ {x \in E } : {x > x _ {0} } \} $, then the right-hand limit of $ f $ at $ x _ {0} $ is simply the infimum of $ A $:

$$ \lim\limits _ {x \downarrow x _ {0} } f ( x) = \inf A . $$

Similar remarks hold for left-hand limits.

How to Cite This Entry:
Increasing function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Increasing_function&oldid=47327
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article