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".