# Function

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)

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.