Is it farther or further?
A function which is not defined for all arguments of its input type. E.g.
f(x) = 1/x if x /= 0.
The opposite of a total function. In denotational semantics, a partial function
f : D -> C
may be represented as a total function
ft : D' -> lift(C)
where D' is a superset of D and
ft x = f x if x in D ft x = bottom otherwise
where lift(C) = C U bottom. Bottom (LaTeX \perp) denotes "undefined".