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FUNCTIONS: SPECIFICATION AND DEFINITION
To deal with functions as a math object, you need a precise definition of "function". That is what this article gives you.
 The article starts by giving a specification of "function".
 After that, we get into the technicalities of the
definitions of the general concept of function.
 Things get complicated because there are several inequivalent definitions of "function" in common use.
Specification of "function"
A function $f$ is a mathematical object which determines and is completely determined by the following data:
 (DOM) $f$ has a domain, which is a set. The domain may be denoted by $\text{dom} f$.
 (COD) $f$ has a codomain, which is also a set and may be denoted by $\text{cod} f$.
 (VAL) For each element $a$ of the domain of $f$, $f$ has a value at $a$.
 (FP) The value of $f$ at $a$ is
completely determined by $a$ and $f$.
 (VIC) The value of $f$ at $a$ must be an element of the codomain of $f$.
 The value of $f$ at $a$ is most commonly written $f(a)$, but see Functions: Notation and Terminology.
 To evaluate $f$ at $a$ means to determine $f(a)$. The two examples of functions below show that different functions may have different strategies for evaluating them.
 In the expression "$f(a)$", $a$ is called the input or (oldfashioned) argument of $f$.
 "FP" means functional property.
 "VIC" means "value in codomain".
Examples
I give two examples here. The examples of functions chapter contains many other examples.
A finite function
Let $F$ be the function defined on the set $\left\{\text{a},\text{b},\text{c},\text{d}\right\}$ as follows: \[F(\text{a})=\text{a},\,\,\,F(\text{b})=\text{c},\,\,\,F(\text{c})=\text{c},\,\,\,F(\text{d})=\text{b}\]In this definition, $\text{a},\text{b},\text{c},\text{d}$ are letters of the alphabet, not variables. This is the function called "Finite'' in the chapter on examples of functions.
 The definition of $F$ says "$F$ is defined on the set $\left\{\text{a},\,\text{b},\,\text{c},\,\text{d} \right\}$". The phrase "is defined on"
means that the domain is that set. That is standard terminology.
 The value of $F$ at each element of the domain is given explicitly. The value at
$\text{b}$, for example, is $\text{c}$, because the definition says that $F(\text{b}) = \text{c}$. No other reason needs to be given. Mathematical definitions can be arbitrary.
 The codomain of $F$ is not specified, but must include the set $\{\text{a},\text{b},\text{c}\}$. The codomain of a function is often not specified when it is not important, which is most of the time in freshman calculus (for example).
 The diagram below shows how $F$ obeys the rule that the value of an element $x$ in the domain is completely determined by $x$ and $F$.
 If two arrows had started from the same element of the domain, then $F$ would not be a function. (It would be a multivalued function).
 If there were an element of the domain that no arrow started from, it $F$ would not be a function. (It would be a partial function.)
 In this example, to evaluate $F$ at $b$ (to determine the value of $F$ at $b$) means to look at the definition of $F$, which says among other things that the value is $c$ (or alternatively, look at the diagram above and see what letter the arrow starting at $b$ points to). In this case, "evaluation" does not imply calculating a formula.
A realvalued function
Let $G$ be the realvalued function defined by the formula $G(x)={{x}^{2}}+2x+5$.
 The definition of $G$ gives the value at each element of the domain by a formula. The value at $3$, for example, is obtained by calculating \[G(3)=3^2+2\cdot3+5=20\]
 The definition of $G$
does not specify the domain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is $\mathbb{R}$.
 The definition of $G$ does not specify the codomain, either. However, the codomain must include all real numbers greater than or equal to $4$. (Why?)
 So if an author wrote, "Let $H(x)=\frac{1}{x}$", the domain would be the set of all real numbers except $0$. But a careful author would write, "Let $H(x)=\frac{1}{x}$ ($x\neq0$)."
What the specification means
 The specification guarantees that a function satisfies all five of the properties listed.
 The specification does not define a mathematical structure in the way mathematical structures have been defined in the past: In particular, it does not require a function to be one or more sets with structure.
 Even so, it is useful to have the specification, because:
Many mathematical definitions introduce extraneous technical elements which clutter up your thinking about the object they define. 
History
The discussion below is an oversimplification of the history of mathematics, which many people have written thick books about. A book relevant to these ideas is Plato's Ghost, by Jeremy Gray.
Until late in the nineteenth century, functions were usually thought of as defined by formulas (including infinite series). Problems arose in the theory of harmonic analysis which made mathematicians require a more general notion of function. They came up with the concept of function as a set of ordered pairs with the functional property (discussed below), and that understanding revolutionized our understanding of math.
In particular, this definition, along with the use of set theory, enabled abstract math (ahem) to become a common tool for understanding math and proving theorems. It is conceivable that some readers may wish it hadn't. Well, tough.
The modern definition
of function given here (which builds on the ordered pairs with functional property definition) came into use beginning in the 1950's. The modern definition became necessary in algebraic topology and is widely used in many fields today.
The concept of function as a formula never disappeared entirely, but was studied mostly by logicians who generalized it to the study of functionasalgorithm. Of course, the study of algorithms is one of the central topics of modern computing science, so the notion of functionasformula (updated to functionasalgorithm) has achieved a new importance in recent years.
To state both the definition, we need a preliminary idea.
The functional property
A set $P$ of ordered pairs has the functional property if two pairs in $P$ with the same first coordinate have to have the same second coordinate (which means they are the same pair). In other words, if $(x,a)$ and $(x,b)$ are both in $P$, then $a=b$.
How to think about the functional property
The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is (which is just what requirement FP says in the specification). That's why you can write "$G(x)$'' for any $x$ in the domain of $G$ and not be ambiguous.
Examples
 The set $\{(1,2), (2,4), (3,2), (5,8)\}$ has the functional property, since no two different pairs have the same first coordinate. Note that there are two different pairs with the same second coordinate. This is irrelevant to the functional property.
 The set $\{(1,2), (2,4), (3,2), (2,8)\}$ does not have the functional property. There are two different pairs with first coordinate 2.
 The empty set $\emptyset$ has the function property vacuously.
Example: graph of a function defined by a formula
In calculus books, a picture like this one (of part of $y=x^2+2x+5$) is called a graph. Here I use the word "graph" to denote the set of ordered pairs
\[\left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{}\,x\in \mathbb{R } \right\}\]
which is a mathematical object rather than some ink on a page or pixels on a screen.
The graph of any function studied in beginning calculus has the functional property. For example, the set of ordered pairs above has the functional property because if $x$ is any real number, the formula ${{x}^{2}}+2x+5$ defines a specific real number.
 if $x = 0$, then ${{x}^{2}}+2x+5=5$, so the pair $(0, 5)$ is an element of the graph of $G$. Each time you plug in $0$ in the formula you get 5.
 if $x = 1$, then ${{x}^{2}}+2x+5=8$.
 if $x = 2$, then ${{x}^{2}}+2x+5=5$.
You can measure where the point $\{2,5\}$ is on the (picture of) the graph and see that it is on the blue curve as it should be.
No other pair whose first coordinate is $2$ is in the graph of $G$, only $(2, 5)$. That is because when you plug $2$ into the formula ${{x}^{2}}+2x+5$, you get $5$ and nothing else. Of course, $(0, 5)$ is in the graph, but that does not contradict the functional property. $(0, 5)$ and $(2, 5)$ have the same second coordinate, but that is OK.
Mathematical definition of function
A function $f$ is a
mathematical structure consisting of the following objects:
 A set called the domain of $f$, denoted by $\text{dom} f$.
 A set called the codomain of $f$, denoted by $\text{cod} f$.
 A set of ordered pairs called the graph of $ f$, with the following properties:
 $\text{dom} f$ \text{dom} fis the set of all first coordinates of pairs in the graph of $f$.
 Every second coordinate of a pair in the graph of $f$ is in $\text{cod} f$ (but $\text{cod} f$ may contain other elements).
 The graph of $f$ has the functional property.
Using
arrow notation, this implies that $f:\text{dom}f\to\text{cod} f$.
Remark
The main difference between the specification of function given previously and this definition is that the definition replaces the statement "$f$ has a value at $a$" by introducing a set of ordered pairs (the graph) with the functional property.
 This set of ordered pairs is extra structure introduced by the definition mainly in order to make the definition a classical setswithstructure.
 This makes the graph, which should be a concept derived from the concept of function, appear to be a necessary part of the function.
 That suggests incorrectly that the graph is more of a primary intuition that other intuitions such as function as map, function as transformer, and other points of view discussed in the article Images and metaphors for functions.
 The concept of graph of a function is indeed an important intuition, and is discussed with examples in the articles Graphs of continuous functions and Graphs of finite functions.
 Nevertheless, the fact that the concept of graph appears in the definition of function does not make it the most important intuition.
Examples
 Let $F$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ and define $A = \{1, 2, 3, 5\}$ and $B = \{2, 4, 8\}$. Then $F:A\to B$ is a function. In speaking, we would usually say, "$F$ is a function from $A$ to $B$."
 Let $G$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ (same as above), and define $A = \{1, 2, 3, 5\}$ and $C = \{2, 4, 8, 9, 11, \pi, 3/2\}$. Then $G:A\to C$ is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in the codomain $C$, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
 Let $H$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Then $H:A\to \mathbb{R}$ is a function, since $2$, $4$ and $8$ are all real numbers.
 Let $D = \{1, 2, 5\}$ and $E = \{1, 2, 3, 4, 5\}$. Then there is no function $D\to A$ and no function $E\to A$ with graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Neither $D$ nor $E$ has exactly the same elements as the first coordinates of the graph.
Identity and inclusion
Suppose we have two setsĀ A andĀ B with $A\subseteq B$.
 The identity function on A is the function ${{\operatorname{id}}_{A}}:A\to A$ defined by ${{\operatorname{id}}_{A}}(x)=x$ for all $x\in A$. (Many authors call it ${{1}_{A}}$).
 When $A\subseteq B$, the inclusion function from $A$ to $B$ is the function $i:A\to B$ defined by $i(x)=x$ for all $x\in A$. Note that there is a different function for each pair of sets $A$ and $B$ for which $A\subseteq B$. Some authors call it ${{i}_{A,\,B}}$ or $\text{in}{{\text{c}}_{A,\,B}}$.
The identity function and an inclusion function for the same set $A$ have exactly the same graph, namely $\left\{ (a,a)a\in A \right\}$.
More about this below.
Other definitions of function
Original abstract definition of function
Definition

A function $f$ is a set of ordered pairs with the functional property.
 If $f$ is a function according to this definition, the domain of $f$ is the set of first coordinates of all the pairs in $f$.
 If $x\in \text{dom} f$, then we define the value of $f$ at $x$, denoted by $f(x)$, to be the second coordinate of the only ordered pair in $f$ whose first coordinate is $x$.
Remarks
Possible confusion
Some confusion can result because of the presence of these two different definitions.

For example, since the identity function ${{\operatorname{id}}_{A}}:A\to A$ and the inclusion function ${{i}_{A,\,B}}:A\to B$ have the same graph, users of the older definition are required in theory to say they are the same function.
 Also it requires you to say that the graph of a function is the same thing as the function.

In my observation, this does not make a problem in practice, unless there is a very picky person in the room.

It also appears to me that the modern definition is (quite rightly) winning and the original abstract definition is disappearing.
Multivalued function
The phrase multivalued function
refers to an object that is like a function $f:S\to T$ except that for $s\in S$, $f(s)$ may denote more than one value.
Examples
 Multivalued functions arose in considering complex functions. In common practice, the symbol $\sqrt{4}$ denoted $2$, although $2$ is also a square root of $4$. But in complex function theory, the square root function takes on both the values $2$ and $2$. This is discussed in detail in Wikipedia.
 The antiderivative is an example of a multivalued operator. For any constant $C$, $\frac{x^3}{3}+C$ is an antiderivative of $x^2$, so that $\frac{x^3}{3}$, $\frac{x^3}{3}+42$, $\frac{x^3}{3}1$ and $\frac{x^3}{3}+2\pi$ are among the infinitely many antiderivatives of $x^2$.
A multivalued function $f:S\to T$ can be modeled as a function with domain $S$ and codomain the set of all subsets of $T$. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two different ways of thinking about
multivalued functions. ("The value may be any of these things..." as opposed to "The value is this whole set of things.")
Some older mathematical papers in complex function theory do not tell you that their functions are multivalued. There was a time when complex function theory was such a Big Deal in research mathematics that the phrase "function theory" meant complex function theory and every mathematician with a Ph. D. knew that complex functions were multivalued.
Partial function
A
partial function
$f:S\to T$ is just like a function except that its input may be defined on only a subset of $S$. For example, the function $f(x):=\frac{1}{x}$ is a partial function from the real numbers to the real numbers.
This models the behavior of computer programs (algorithms): if you consider a program with one input and one output as a function, it may not be defined on some inputs because for them it runs forever (or gives an error message).
In some texts in computing science and mathematical logic, a function is by
convention
a partial function, and this fact may not be mentioned explicitly, especially in research papers.
The phrases "multivalued function" and "partial function" upset some picky types who say things like, "But a multivalued function is not a function!". A hot dog is not a dog, either. I once had a Russian teacher who was Polish and a German teacher who was Hungarian. So what? See the Handbook (click on
radial category).
New approaches to functions
All the definitions of function given here produce
mathematical structures, using the traditional way to define mathematical objects in terms of sets. Such definitions have disadvantages.
Mathematicians have many ways to think about functions.
That a function is a set of ordered pairs with a certain property (functional) and possibly some ancillary ideas (domain, codomain, and others) is not the way we usually think about them$\ldots$Except when we need to reduce the thing we are studying to its absolutely most abstract form to make sure our proofs are correct.
That most abstract form is what I have called the rigorous view or the dry bones and it is when that reasoning is needed that the setswithstructure approach has succeeded.
Our practice of abstraction has led us to new approaches to talking about functions. The most important one currently is category theory. Roughly, a category is a bunch of objects together with some arrows going between them that can be composed head to tail. Functions between sets are examples of this: the sets are the objects and the functions the arrows. But arrows in a category do not have to be functions; in that way category theory is an abstraction of functions.
This abstracts the idea of function in a way that brings out common ideas in various branches of math. Research papers in many branches of mathematics now routinely use the language of category theory. Categories now appear in some undergraduate math courses, meaning that Someone needs to write a chapter on category theory for abstractmath.org.
Besides category theory, computing scientists have come up with other abstract ways of dealing with functions, for example type theory. It has not come as far along as category theory, but has shown recent signs of major progress.
Both category theory and type theory define math objects in terms of their effect on and relationship with other math objects. This makes it possible to do abstract math entirely without using setswithstructure as a means of defining concepts.
References
 Functions in Wikipedia. This is an extensive and mostly welldone description of the use of functions in mathematics.
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