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From The Free On-line Dictionary of Computing (05 January 2017) [foldoc]:
function
mapping
1. (Or "map", "mapping") If D and C are sets
(the domain and codomain) then a function f from D to C,
normally written "f : D -> C" is a subset of D x C such that:
1. For each d in D there exists some c in C such that (d,c) is
an element of f. I.e. the function is defined for every
element of D.
2. For each d in D, c1 and c2 in C, if both (d,c1) and (d,c2)
are elements of f then c1 = c2. I.e. the function is uniquely
defined for every element of D.
See also image, inverse, partial function.
2. Computing usage derives from the mathematical
term but is much less strict. In programming (except in
functional programming), a function may return different
values each time it is called with the same argument values
and may have side effects.
A procedure is a function which returns no value but has
only side-effects. The C language, for example, has no
procedures, only functions. ANSI C even defines a type,
void, for the result of a function that has no result.
(1996-09-01)