# Partial function

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Definition: Partial function
A partial function is a subset (called

the support of ), along with a function We say that if and

is undefined if .

A partial function is total if the support is equal to the domain, i.e. if is a function.

Note that, somewhat confusingly, not all partial functions are functions. However, we do consider a (total) function to be a partial function. We use this terminology because it is standard in mathematics and because it simplifies proofs (so that we don't have to say "a partial or total function" in places where the same argument works for both).

To summarize,

is a partial function, but not a function (because it is not total), while

is a partial function, a function and a total function.