THC Science

In mathematics, a function^[1] is originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of the time. Historically, the concept was elaborated with the infinitesimal calculus at the end of 17th century, and, until the 19th century, the function that were considered were differentiable (that is they have a high degree of regularity). The concept of function has been formalized at the end of the 19th century in terms of set theory, and this has much enlarged the domains of application of the concept.

A function is a process or a relation that associates each element $x$ of a set $X$ , the domain of the function, to a unique element $y$ of another set $Y$ (possibly the same set), the codomain of the function. If the function is called $f$ , this relation is denoted $y = f (x)$ (read $f$ of $x$ ), the element $x$ is the argument or input of the function, and $y$ is the value of the function, the output, or the image of $x$ by $f$ . The symbol that is used for representing the input is the variable of the function (one says often that $f$ is a function of the variable $x$ ).^[2]

A function is uniquely represented by its graph which is the set of all pairs $(x, f (x))$ . When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere, for example in newspapers.

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said the they are "the central objects of investigation" in most fields of mathematics.^[3]

Definition

The above diagram represents a function with domain {1, 2, 3}, codomain {A, B, C, D} and set of ordered pairs {(1,D), (2,C), (3,C)}. The image is {C,D}.

However, this second diagram does not represent a function. One reason is that 2 is the first element in more than one ordered pair. In particular, (2, B) and (2, C) are both elements of the set of ordered pairs. Another reason, sufficient by itself, is that 3 is not the first element (input) for any ordered pair. A third reason, likewise, is that 4 is not the first element of any ordered pair.

When students first begin to study functions, intuitive ideas such as "rules" and "associates" are used. In more advanced mathematics, more precise definitions are needed. The precise mathematical definition of a function relies on the notion of an ordered pair.

A function $f$ from $X$ to $Y$ is defined by a set of ordered pairs $(x, y)$ such that $x \in X$ and $y \in Y$ , which is subject to the following condition: every element of $X$ is the first component of one and only one ordered pair in the set.^[4] In other words, for every $x$ in $X$ there is exactly one element $y$ such that the ordered pair $(x, y)$ belongs to the set of pairs defining the function $f$ .

In this definition, $X$ and $Y$ are respectively called the domain and the codomain of the function $f$ . If $(x, y)$ belongs to the set defining $f$ , then $y$ is the value of $f$ for the argument $x$ , or the image of $x$ under $f$ . One says also that $y$ is the value of $f$ for the value $x$ of the variable.

The range denotes generally the image of a function, that is the set of all images of elements of the domain. However it is sometimes used as a synonym of codomain, generally in old textbooks.

Notation

A function $f$ with domain $X$ and codomain $Y$ is commonly denoted

f\colon X\to Y

or

X~{\stackrel {f}{\to }}~Y.

The notation

y=f(x)

means that $(x, y)$ belongs to the set of pairs defining the function $f$ . When it is needed to refer explicitly to this set of pairs, it is denoted

\{(x,f(x))\colon x\in X\}.

The symbol used to denote a particular function consists generally of a single letter in italic font, most often the lower-case letters $f, g, h$ . Some widely used functions are represented by a symbol consisting of several letters (generally an abbreviation of their name). In this case, it is set in roman type, such as " $sin$ " for the sine function.

Sometimes the parentheses of the functional notation are omitted if no ambiguity arises, as for $\sin x$ instead of $\sin(x).$

In many cases, a function is defined by a formula that describes how the value of the function is computed from the value of the argument. For example, if a multiplication is defined on a set $X$ , then one may define a square function with domain and codomain $X$ by

x\mapsto x\cdot x,

or, more commonly,

x\mapsto x^{2}.

Similarly, the identity function on a set $X$ is the function

x\mapsto x.

This notation $\mapsto$ is useful for avoiding naming a function when not necessary, as in the sentence "the square function is the function $x\mapsto x^{2}.$

The definition of a function by a formula may use case distinction as for the definition of the absolute value of real numbers:

x\mapsto |x|=\left\{{\begin{array}{rl}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{array}}\right.

Normally the definition of a function includes the definition of its domain and its codomain, but it is not always the case. Typically, a function from the reals to the reals does not always refer to a function that has the real numbers a domain. This refers often to a function which has a subset of the reals as domain. In many cases some computation is needed if one want to know the domain exactly (see partial function). For example, if $f$ is a function that has the real numbers as domain and codomain, then $x\mapsto {\frac {1}{f(x)}}$ is a (possibly partial) function from the reals to the reals whose domain is the set of the reals $x$ such that $f (x) \neq 0$ . In many cases, this set is difficult to determine exactly, and often, this is not a problem for working with the function.

When the domain of a function is the set of natural numbers, one talks often of a sequence (mathematics), and one uses the index notation, that is, one denotes

f_{n}

the image of the nonnegative integer $n$ by the function or sequence $f$ . One says also that $f_{n}$ is the $n$ th element of the sequence.

Finally, dot notation is occasionally used for avoiding naming the variable. For example, $a(\cdot )^{2}$ may stand for $x\mapsto ax^{2}$ , and $\textstyle \int _{a}^{\,(\cdot )}f(u)du$ may stand for the integral function $\textstyle x\mapsto \int _{a}^{x}f(u)du$ .

Specifying a function

In general, a specific function is defined by associating to every element of the domain a property of this element (as in the above example of colored shapes) or the result of a computation taking as input. This computation may be described by a formula. (This is the starting point of algebra of replacing many similar computations on numbers by a single formula that describes the computation on variables representing unspecified numbers). This definition of a function uses frequently previously defined auxiliary functions.

For example, the function $f$ from the reals to the reals, defined by $f(x)={\sqrt {1+x^{2}}},$ use the square function and the square root function as auxiliary functions. The domain of $f$ is the whole set of real numbers, even if domain of the square root function is restricted to the positive real numbers.

The computation that allows defining a function may often be described by an algorithm, and any kind of algorithm may be used. Sometimes, the definition of a function may involve elements or properties that can be defined, but not computed. For example, if one, consider the set ${\mathcal {P}}$ of the programs in a given programming language, which take an integer as input. The terminating function is the function that returns 1 if a progam of ${\mathcal {P}}$ runs for ever when executed on a given integer input, and returns 0 otherwise. It is a basic theorem of computability theory that it cannot exist an algorithm for computing this function.

More generally, the computability theory is the study of the computable functions, that is the functions that can be computed by an algorithm.

Above ways of defining functions define them "pointwise", that is, each value is defined independently of the other values. This is not always the case.

When the domain of a function is the nonnegative integers or, more generally, when the domain may be well ordered, a function may be defined by induction or recursion, which means (roughly) that the definition of the function for a given input depends on previously defined values of the function. For example, the Fibonacci sequence is a function from the natural numbers into themselves that is defined by (see above for the use of indices for the argument of a function)

F_{0}=0,F_{1}=1,{\text{ and }}F_{n}=F_{n-1}+F_{n-2}{\text{ for }}n>1.

In calculus, the functions that are considered have some regularity. That is, the value of the function at a point is related with the values of the function at neighbor points. This allows defining them by functional equations (the gamma function is the unique meromorphic function such that $\Gamma (z+1)=z\Gamma (z)$ and $\Gamma (1)=1$ ), by differential equations (the natural logarithm is the solution of the differential equation ${\frac {\partial \ln x}{\partial x}}={\frac {1}{x}}$ such that $ln(1) = 0$ ), integral equations or analytic continuation.

Representing a function

As functions may be complicated objects, it is often useful to draw the graph of a function for getting a global view of its properties. Some functions may also represented histograms

Graph

Given a function $f:X\to Y,$ its graph is, formally, the set

G=\{(x,y):x\in X{\text{ and }}y=f(x)\}.

In the frequent case where $X$ and $Y$ are subsets of the real numbers (or may be identified to such subsets), an element $(x,y)\in G$ may be identified with the point of coordinates $x, y$ in the Cartesian plane. Marking these points provides a drawing, generally a curve, that is also called the graph of the function. For example the graph of the square function

x\mapsto x^{2}

is a parabola that consists of all points of coordinates $(x,x^{2})$ for $x\in \mathbb {R} .$

It is possible to draw effectively the graph of a function only if the function is sufficiently regular, that is, either if the function is differentiable (or piecewise differentiable) or if its domain may be identified with the integers or a subset of the integers.

If either the domain or the codomain of the function is a subset of $\mathbb {R} ^{n},$ the graph is a subset of a Cartesian space of higher dimension, and various technics have been developed for drawing it, including the use of colors for representing one of the dimensions.^{[citation needed]}

Histogram

Histograms are often used for representing functions whose domain is finite, or is the natural numbers or the integers. In this case, an element $x$ of the domain is represented by an interval of the $x$ -axis, and a point $(x, y)$ of the graph is represented by a rectangle with basis the interval corresponding to $x$ and height $y$ .

In statistic, histogram are often used for representing very irregular functions. For example, for representing the function that associates his weight to each member of some population, one draws the histogram of the function that associates to each weight interval the number of people, whose weights belong to this interval.

There are many variants of this method, see Histogram for details.

General properties

In this section, we describe general properties of functions, that are independent of specific properties of the domain and the codomain.

Canonical functions

Some functions are uniquely defined by their domain and codomain, and are sometimes called canonical:

For every set $X$ , there is a unique function, called the empty function from the empty set to $X$ . This function is not interesting by itself, but useful for simplifying statements, similarly as the empty sum (equal to 0) and the empty product equal to 1.
For every set $X$ and every singleton set ${s}$ , there is a unique function, called the canonical surjection, from $X$ to ${s}$ , which maps to $s$ every element of $X$ . This is a surjection (see below), except if $X$ is the empty set.
Given a function $f\colon X\to Y,$ the canonical surjection of $f$ onto its image $f(X)=\{f(x)\mid x\in X\}$ is the function from $X$ to $f (X)$ that maps $x$ to $f (x)$
For every subset $X$ of a set $Y$ , the canonical injection of $X$ into $Y$ is the injective (see below) function that maps every element of $X$ to itself.
The identity function of $X$ , often denoted by $\operatorname {id} _{X}$ is the canonical injection of $X$ into itself.

Function composition

Given two functions $f:X\to Y$ and $g:Y\to Z$ such that the domain of $g$ is the codomain of $f$ , their composition is the function $g\circ f\colon X\rightarrow Z$ defined by

(g\circ f)(x)=g(f(x)).

That is, the value of $g\circ f$ is obtained by first applying $f$ to $x$ to obtain $y = f (x)$ and then applying $g$ to the result $y$ to obtain $g (y) = g (f (x))$ . In the notation the function that is applied first is always written on the right.

The composition $g\circ f$ is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when $g\circ f$ and $f\circ g$ are both defined, the composition is not commutative. For example, if $f (x) = x 2$ and $g (x) = x + 1$ , one has $g(f(x))=x^{2}+1\neq f(g(x))=(x+1)^{2}.$

The function composition is associative in the sense that, if one of $(h\circ g)\circ f$ and $h\circ (g\circ f)$ is defined, then the other is also defined, and they are equal. Thus, one writes

h\circ g\circ f=(h\circ g)\circ f=h\circ (g\circ f).

The identity functions $\operatorname {id} _{X}$ and $\operatorname {id} _{Y}$ are respectively a right identity and a left identity for functions from $X$ to $Y$ . That is, if $f$ is a function with domain $X$ , and codomain $Y$ , one has $f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.$

A composite function g(f(x)) can be visualized as the combination of two "machines".
A simple example of a function composition
Another composition. For example, we have here $(g \circ f)(c) = #$ .

Image and preimage

Let $f\colon X\to Y.$ The image by $f$ of an element $x$ of the domain $X$ is $f (x)$ . If $A$ is any subset of $X$ , then the image of $A$ by $f$ , denoted $f (A)$ is the subset of the codomain $Y$ consisting of all images of elements of $A$ , that is,

f(A)=\{f(x)\mid x\in A\}.

The image of $f$ is the image of the whole domain, that is $f (X)$ . It is also called the range of $f$ , although the term may also refer to the codomain.^[5]

On the other hand, the inverse image, or preimage by $f$ of a subset $B$ of the codomain $Y$ is the subset of the domain $X$ consisting of all elements of $X$ whose images belong to $B$ . It is denoted by $f^{-1}(B).$ That is

f^{-1}(B)=\{x\in X\mid f(x)\in B\}.

For example, the preimage of {4, 9} under the square function is the set {−3,−2,2,3}.

By definition of a function, the image of an element $x$ of the domain is always a single element of the codomain. However, the preimage of a single element $y$ , denoted $f^{-1}(x),$ may be empty or contain any number of elements. For example, if $f$ is the function from the integers to themselves that map every integer to 0, then $f -1 (0) = Z$ .

If $f\colon X\to Y$ is a function, $A$ and $B$ are subsets of $X$ , and $C$ and $D$ are subsets of $Y$ , then one has the following properties:

$A\subseteq B\Longrightarrow f(A)\subseteq f(B)$
$C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)$
$A\subseteq f^{-1}(f(A))$
$C\supseteq f(f^{-1}(C))$
$f(f^{-1}(f(A)))=f(A)$
$f^{-1}(f(f^{-1}(C)))=f^{-1}(C)$

The preimage by $f$ of an element $y$ of the codomain is sometimes called, in some contexts, the fiber of $y$ under $f$ .

If a function $f$ has an inverse (see below), this inverse is denoted $f^{-1}.$ In this case $f^{-1}(C)$ may denote either the image by $f^{-1}$ or the preimage by $f$ of $C$ . This is not a problem, as these sets are equal. The notation $f(A)$ and $f^{-1}(C)$ may be ambiguous in the case of sets that contain some subsets as elements, such as $\{x,\{x\}\}.$ In this case, some care may be needed, for example, by using square brackets $f[A],f^{-1}[C]$ for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

Injective, surjective and bijective functions

Let $f\colon X\to Y$ be a function.

The function $f$ is injective (or one-to-one, or is an injection) if $f (a) \neq f (b)$ for any two different elements $a$ and $b$ of $X$ . Equivalently, $f$ is injective if, for any $y\in Y,$ the preimage $f^{-1}(y)$ contains at most one element. An empty function is always injective. If $X$ is not the empty set, and if, as usual, the axiom of choice is assumed, then $f$ is injective if and only if there exists a function $g\colon y\to X$ such that $g\circ f=\operatorname {id} _{X},$ that is, if $f$ has a left inverse. The axiom of choice is needed, because, if $f$ is injective, one defines $g$ by $g(y)=x$ if $y=f(x),$ and by $g(y)=x_{0}$ , if $y\not \in f(X),$ where $x_{0}$ is an arbitrarily chosen element of $X$ .

The function $f$ is surjective (or onto, or is a surjection) if the range is equals the codomain, that is, if $f (X) = Y$ . In other words, the preimage $f^{-1}(y)$ of every $y\in Y$ is nonempty. If, as usual, the axiom of choice is assumed, then $f$ is surjective if and only if there exists a function $g\colon y\to X$ such that $f\circ g=\operatorname {id} _{Y},$ that is, if $f$ has a right inverse. The axiom of choice is needed, because, if $f$ is injective, one defines $g$ by $g(y)=x,$ where $x$ is an arbitrarily chosen element of $f^{-1}(y).$

The function $f$ is bijective (or is bijection or a one-to-one correspondence) if it is both injective and surjective. That is $f$ is bijective if, for any $y\in Y,$ the preimage $f^{-1}(y)$ contains exactly one element. The function $f$ is bijective if and only if it admits an inverse function, that is a function $g\colon y\to X$ such that $g\circ f=\operatorname {id} _{X},$ and $f\circ g=\operatorname {id} _{Y}.$ (Contrarily to the case of injections and surjections, this does not require the axiom of choice.)

Every function $f\colon X\to Y$ may be factorized as the composition $i \circ s$ of a surjection followed by an injection, where $s$ is the canonical surjection of $X$ onto $f (X)$ , and $i$ is the canonical injection of $f (X)$ into $Y$ . This is the canonical factorization of $f$ .

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " $f$ maps $X$ onto $Y$ " differs from " $f$ maps $X$ into $B$ " in that the former implies that $f$ is surjective), while the latter makes no assertion about the nature of $f$ the mapping. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage to be more symmetrical.

Restriction and extension

If $f\colon X\to Y$ is a function, and $S$ is a subset of $X$ , then the restriction of $f$ to $S$ , denoted $f | S$ , is the function from $S$ to $Y$ that is defined by

f_{|S}(x)=f(x)\quad {\text{for all }}x\in S.

This often used for define partial inverse functions: if there is a subset $S$ of a function $f$ such that $f | S$ is injective, then the canonical surjection of $f | S$ on its image $f | S (S) = f (S)$ is a bijection, which has an inverse function from $f (S)$ to $S$ . This is in this way that inverse trigonometric functions are defined. The cosine function, for example, is injective, when restricted to the interval $(-0, π)$ ; the image of this restriction is the interval $(-1, 1)$ ; this defines thus an inverse function from $(-1, 1)$ to $(-0, π)$ , which is called arccosine and denoted $arccos$ .

Function restriction may also be used for "gluing" functions together: let $\textstyle X=\bigcup _{i\in I}U_{i}$ be the decomposition of $X$ as a union of subsets. Suppose that a function $f_{i}\colon U_{i}\to Y$ is defined on each $U_{i},$ such that, for each pair of indices, the restrictions of $f_{i}$ and $f_{j}$ to $U_{i}\cap U_{j}$ are equal. Then, this defines a unique function $f\colon X\to Y$ such that $f_{|U_{i}}=f_{i}$ for every $i$ . This is generally in this way that functions on manifolds are defined.

An extension of a function $f$ is a function $g$ such that $f$ is a restriction of $g$ . A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying homographies of the real line. An homography is a function $h(x)={\frac {ax+b}{cx+d}}$ such that $ad - bc \neq 0$ . Its domain is the set of all real numbers different from $-d/c,$ and its image is the set of all real numbers different from $a/b.$ If one extends the real line to the projectively extended real line by adding $\infty$ to the real numbers, one may extend $h$ for being a bijection of the extended real line to itself, by setting $h(\infty )=a/b$ and $h(-d/c)=\infty .$

Multivariate function

A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time and its speed.

More formally, a function of $n$ variables is a function whose domain is a set of $n$ -tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain it the set of all pairs (2-uples) of integers and the codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function.

The direct product $X_{1}\times \cdots \times X_{n}$ of $n$ sets $X_{1},\ldots ,X_{n}$ is the set of all $n$ -uples $(x_{1},\ldots ,x_{n})$ such that $x_{i}\in X_{i}$ for every $i$ . Therefore, a function of $n$ variables is a function

f:U\to Y,

where the domain $U$ has the form

U\subseteq X_{1}\times \cdots \times X_{n}.

When using the functional notation, one omits the parentheses surrounding tuples. That is, one write $f(x_{1},x_{2})$ , and not $f((x_{1},x_{2})).$

In the case where all $X_{i}$ s are equal to the set $\mathbb {R}$ of real numbers, one has a function of several real variables. If the $X_{i}$ s are equal to the set $\mathbb {C}$ of complex numbers, one has a function of several complex variables

It is common to consider also functions whose codomain is a product of sets. For example, Euclidean division maps very pair $(a, b)$ of integers with $b \neq 0$ to a pair of integers called the quotient and the remainder.

{\begin{aligned}{\text{Euclidean division}}\colon &\quad \mathbb {Z} \times (\mathbb {Z} \setminus \{0\})\to \mathbb {Z} \times \mathbb {Z} \\&(a,b)\to (\operatorname {quotient} (a,b),\operatorname {remainder} (a,b)).\end{aligned}}

The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field.

In calculus

The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus (see History of the function concept). At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to function of a complex variable. In the second half of 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.

Real function

Graph of a polynomial function, here a quadratic function.

A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.

Functions enjoy pointwise operations, that is, if $f$ and $g$ are functions, their sum, difference and product are functions defined by

{\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.

The domains of the resulting functions are the intersection of the domains of $f$ and $g$ . The quotient of two functions is defined similarly by

{\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},

but the domain of the resulting function is obtained by removing the zeros of $g$ from the intersection of the domains of $f$ and $g$ .

The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function $x\mapsto {\frac {1}{x}},$ whose graph is an hyperbola, and whose domain is the whole real line except for 0.

The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that is differentiable in any open interval in which the original function is continuous. For example, the function $x\mapsto {\frac {1}{x}}$ is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for $x = 1$ , is a differentiable function called the natural logarithm.

A real function $f$ is monotonic in an interval if the sign of ${\frac {f(x)-f(y)}{x-y}}$ does not depend of the choice of $x$ and $y$ in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function $f$ is monotonic in an interval $I$ , it has an inverse function, which is a real function with domain $f (I)$ and image $I$ . This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.

Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example the sine and the cosine functions are the solutions of the linear differential equation

y''+y=0

such that

\sin 0=0,\quad \cos 0=1,\quad {\frac {\partial \sin x}{\partial x}}(0)=1,\quad {\frac {\partial \cos x}{\partial x}}(0)=0.

Complex function

Function of several real or complex variables

Vector-valued function

Function space

In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions.

Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.

Multi-valued functions

Several methods for specifying functions of real or complex variables start from a local definition of the funcion at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point $x_{0},$ there are several possible starting values for the function.

For example, for defining the square root function as the inverse function of the square function, one may choose any positive real number for $x_{0},$ and there are two choices for the value of the function, one which is positive and denoted ${\sqrt {x_{0}}},$ and the other that is denoted $-{\sqrt {x_{0}}}.$ These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive $x$ , one value for 0 and no value for negative $x$ .

In the preceding example, one choice, the positive square root, is more natural then the other. This is not the case in general. For example, let consider the implicit function that maps $y$ to a root $x$ of $x^{3}-3x-y=0$ (see the figure on the right). For $y = 0$ one may choose either $0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}$ for $x.$ By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval $[-2, 2]$ and the image is $[-1, 1]$ ; for the second one, the domain is $[-2, +\infty)$ and the image is $[1, +\infty)$ ; for the last one, the domain is $(-\infty, 2]$ and the image is $(-\infty, -1]$ . As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of $y$ that has three values for $-2 < y < 2$ , and only one value for $y \leq -2$ and $y \geq -2$ .

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function is analytic may be extended by analytic continuation consists generally of almost the whole complex plane. However, it is frequent that, when extending the domain through two different paths, one gets different values. For example, when extending the domaine of the square root function, along a path of complex numbers with positive imaginary parts, one gets $i$ for the square root of –1, while, when extending through complex numbers with negative imaginary parts, one gets $- i$ . There are generally two ways for solving the problem. One may define a function that is not continuous along some curve, called branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.

In foundations of mathematics and set theory

The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is general not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.

For example the singleton set may be considered as a function $x\mapsto \{x\}.$ Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.^[6]

These generalized functions may be critical in the development of a formalization of foundations of mathematics. For example, the Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be interpreted as "if $X$ is a set, and $F$ is a function, then $F [X]$ is a set".

In computer science

^ The words map or mapping, transformation, correspondence, and operator are often used synonymously. Halmos 1970, p. 30.
^ MacLane, Saunders; Birkhoff, Garrett (1967). Algebra (First ed.). New York: Macmillan. pp. 1–13.
^ Spivak 2008, p. 39.
^ Hamilton, A. G. Numbers, sets, and axioms: the apparatus of mathematics. Cambridge University Press. p. 83. ISBN 0-521-24509-5.
^ Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, page 15. ISO 80000-2 (ISO/IEC 2009-12-01)
^ Gödel 1940, p. 16; Jech 2003, p. 11; Cunningham 2016, p. 57

References

Bartle, Robert (1967). The Elements of Real Analysis. John Wiley & Sons. {{cite book}}: Invalid |ref=harv (help)
Bloch, Ethan D. (2011). Proofs and Fundamentals: A First Course in Abstract Mathematics. Springer. ISBN 978-1-4419-7126-5. {{cite book}}: Invalid |ref=harv (help)
Cunningham, Daniel W. (2016). Set theory: A First Course. Cambridge University Press. ISBN 978-1107120327. {{cite book}}: Invalid |ref=harv (help)
Gödel, Kurt (1940). The Consistency of the Continuum Hypothesis. Princeton University Press. ISBN 978-0691079271. {{cite book}}: Invalid |ref=harv (help)
Halmos, Paul R. (1970). Naive Set Theory. Springer-Verlag. ISBN 0-387-90092-6. {{cite book}}: Invalid |ref=harv (help)
Jech, Thomas (2003). Set theory (Third Millennium ed.). Springer-Verlag. ISBN 3-540-44085-2. {{cite book}}: Invalid |ref=harv (help)
Spivak, Michael (2008). Calculus (4th ed.). Publish or Perish. ISBN 978-0-914098-91-1. {{cite book}}: Invalid |ref=harv (help)

External links

"Function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Function". MathWorld.
The Wolfram Functions Site gives formulae and visualizations of many mathematical functions.
NIST Digital Library of Mathematical Functions

[1] The words map or mapping, transformation, correspondence, and operator are often used synonymously. Halmos 1970, p. 30.

[MacLane-2] MacLane, Saunders; Birkhoff, Garrett (1967). Algebra (First ed.). New York: Macmillan. pp. 1–13.

[FOOTNOTESpivak200839-3] Spivak 2008, p. 39.

[4] Hamilton, A. G. Numbers, sets, and axioms: the apparatus of mathematics. Cambridge University Press. p. 83. ISBN 0-521-24509-5.

[standard-5] Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, page 15. ISO 80000-2 (ISO/IEC 2009-12-01)

[6] Gödel 1940, p. 16; Jech 2003, p. 11; Cunningham 2016, p. 57

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