A partial function is like a function from to except that may not be defined for every element of . (Compare a multi-valued function, where may have several possible values.)
In some fields (including secondary-school mathematics even today), functions are often considered to be parital by default, requiring one to specify a total function otherwise. As category-theoretic and type-theoretic formalisation spreads, this is difficult to treat as the basic concept, and the most modern idea is that a function must be total. If you want partial functions, then you can get them in terms of total functions as below.
Given sets and , a partial function from a to consists a subset of and a (total) function from to . In more detail, this is a span
of total functions, where is an injection. (This condition can be dropped to define a partial multi-valued function, which is simply a span.)
and are called the source and target of as usual; then is the domain of and is the inclusion of the domain into the source. By abuse of notation, the partial function is conflated with the (total) function .
Notice that the induced function is an injection, so a partial function is the same as a functional relation seen from a different point of view.
We consider two partial functions (with the same given source and target) to be equal if there is a bijection between their domains that makes the obvious diagrams commute. You can get this automatically in a traditional set theory by requiring to be literally a subset of (with the inclusion map).
In secondary-school mathematics, one often makes functions partial by fiat, just to see if students can calculate the domains of composite functions and the like. This is not (only) busywork, as in applications one often has a function given by a formula that is really valid only on a certain domain. However, in more sophisticated analyses (such as those that Lawvere and his followers propose for physics and synthetic geometry), these domains and the total functions on them become the primary objects of study, with the partial functions being secondary (as is seen as merely a way to place coordinates on ).
In analysis, one often considers partial functions whose domains are required to be intervals in the real line, regions of the complex plane, or dense subsets of a Banach space.
Ronnie There is an interesting debate possible here!
On the basis of my teaching of first year analysis and calculus since 1959, I found the most convenient idea is that of a function being a partial function with a domain which can be calculated from a formula for the function, and may be empty. Then one finds that the inverse of an injective function is also a function . A first order differential equation has a solution which is a partial function. What seems to be lacking is the functional analysis of such solutions. For example has a solution whose domain varies with and ought to (and can be made to) vary continuously, including its open domain!
The work of Charles Ehresmann is full of partial functions, derived from his strong interest in analysis and differential geometry, and local-to-global problems. So he developed for example the theory of pseudogroups, and contributed to inverse semigroups.
A possible reason for the difficulties some have of accepting groupoids rather than groups is that groupoids have a partial composition, which is of course very intuitive when one thinks of composing journeys.
In higher dimensional algebra one is dealing with algebraic structures whose domains are defined by geometric conditions.
Of course category theory initially derived from algebra and algebraic topology, where partial functions are unusual. However they are necessary in dealing with fibred exponential laws, i.e. exponential laws in a slice category of , and their applications. See papers of Peter Booth.
Toby: For first-year calculus, I agree with you, except that you ought to be able to restrict the definition to an interval (or a union of intervals) by fiat. (Actually, you can get this from formulas by adding appropriate terms of the form , , etc, but that's silly.) So calculus is about (certain) functions to from unions of intervals on . Of course, this doesn't include all partial functions on , but then it doesn't even include all such total functions, so maybe the restriction on allowed domains doesn't matter.
But if you disagree that ‘in more sophisticated analyses …, these domains and the total functions on them become the primary objects of study’, then feel free to change the text (say to ‘in other analyses …’; I don't intend to defend the claim that this is really the right way to do things.
In a field, the multiplicative inverse is a partial function whose domain is the set of non-zero elements of the field.
The category of sets and partial functions between them is important for understanding computation. However, one often replaces this with an equivalent category of sets and total functions.
Specifically, replace each set with the set of all subsets of with at most one element. In this context, we identify an element of with the subset and write the empty subset as . Then a partial functon becomes a total function such that inhabited subsets of are assigned only to inhabited subsets of . Then is equivalent to the category of such sets and functions.
Classically, , although this is not true constructively. Then the category becomes equivalent to the category of pointed sets and total point-preserving functions. Traditionally, one uses the notation of but (unless one is a constructivist) thinks of this as simply different notation for .
For a more sophisticated analysis of computation, can be replaced with a suitable category of domains, such as directed complete partially ordered sets (DCPOs). The requirement that be preserved can then be removed to model lazy computation, but now we are hardly talking about partial functions anymore.