The Free On-line Dictionary of Computing (30 December 2018):

partial function

   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".

   (1995-02-03)