natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
In a strict sense of the term, a function is a homomorphism $f : S \to T$ of sets. We may also speak of a map or mapping, but those terms are used in other ways in other contexts.
A function from a set $A$ to a set $B$ is determined by giving, for each element of $A$, a specified element of $B$. The process of passing from elements of $A$ to elements of $B$ is called function application. The set $A$ is called the domain of $f$, and $B$ is called its codomain.
A function is sometimes called a total function to distinguish it from a partial function.
More generally, every morphism between objects in a category may be thought of as a function in a generalized sense. This generalized use of the word is wide spread (and justified) in type theory, where for $S$ and $T$ two types, there is a function type denoted $S \to T$ and then the expression $f : S \to T$ means that $f$ is a term of function type, hence is a function.
In this generalized sense, functions between sets are the morphisms in the category Set. This is cartesian closed, and the function type $S \to T$ is then the function set.
For more on this more general use of “function” see at function type.
The formal definition of a function depends on the foundations chosen.
In material set theory, a function $f$ is often defined to be a set of ordered pairs such that for every $x$, there is at most one $y$ such that $(x,y)\in f$. The domain of $f$ is then the set of all $x$ for which there exists some such $y$. This definition is not entirely satisfactory since it does not determine the codomain (since not every element of the codomain may be in the image); thus to be completely precise it is better to define a function to be an ordered triple $(f,A,B)$ where $A$ is the domain and $B$ the codomain.
In structural set theory, the role of functions depends on the particular axiomatization chosen. In ETCS, functions are among the undefined things, whereas in SEAR, functions are defined to be particular relations (which in turn are undefined things).
In type theory, functions are simply terms belonging to function types.
See set theory and type theory for more details.
If we regard sets as discrete categories, then a function is a functor between sets. The functoriality structure becomes the property that a function preserves equality:
One can also speak of functions between proper classes, although the precise details may vary depending on the status of classes with respect to the formal theory.
In ZFC for example, proper classes are by design not formal objects in the theory; rather they are proxied by a formula in the language (for instance, the class $V$ of all sets is proxied by the formula $x = x$; intuitively we may think of $V$ as consisting of all $x$ satisfying that formula). Then functions $f: A \to B$ between classes are again classes given by suitable formulas; see for example the MathOverflow discussion what-are-maps-between-proper-classes. If (as described above for material set theory) one wants to describe a function as an ordered triple $(A, f, B)$, then this too can be accommodated if one defines ordered triples/pairs of classes appropriately; see here for one possibility. Thus functors between categories whose objects and morphisms form proper classes can similarly be described in the language.
Such technical hacks can be avoided by choosing a different foundations. For example, Mac Lane in his Categories for the Working Mathematician assumes ZFC with a universe in which some sets are considered large, such as the set of small sets, so that a category like $Set$ (the category of small sets) is again a formal object of the theory.
Useful terms, more or less synonymous with function, are assignment, assignation or more specifically assignation on objects. These do not have standard meanings but are useful to signal to readers that the domain of the ‘function’ under consideration is large, or that one is more interested in functorial extensions of this partial assignation (cf. e.g. Richard Garner, Homomorphisms of higher categories, Adv. Math. 224 (2010) 2269-2311 for many examples). In mathematical writing “assignment” is usually synonymous with function or map or “mapping”. For example one might speak of “assigning to each positive number its square root” to refer to the function $\sqrt{(-)} \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$.
Authors may resort to verb forms such as “assigns” or “associates” or “sends” in informal writing, perhaps to avoid the bother of specifying an axiomatic framework in which a formal notion like “function” is ensconced. For example, according to Wikipedia, Jacobson defines a functor $F: C \to D$ between categories as a “mapping” that “associates” to each object $X$ in $C$ an object $F(X)$ in $D$, etc. No clarity would be gained by making this any more formal (which as we saw in the case of functions between classes, such as classes of objects of categories, may involve annoyingly technical hacks).
Sometimes the word “assignment” is understood more generally as relation, often when authors define a function to be something that “assigns unique values” (for instance here).
Many functions that carry special names map some ring or field like the real numbers or complex numbers to itself:
…
Special properties these may have:
…