Partial function

In mathematics and theoretical computer science, a partial function on a set is a function whose domain of definition need not be the whole set.

We may define a partial function f from X to Y by extending the codomain Y to Y* by an element ω for "undefined", assumed not to be in Y, so that f is a function in the usual sense from X to Y*, and we regard the domain of definition of f as the subset of X on which f does not take the value ω.

A function on X is total if its domain of definition is the whole of f: that is, if f is a function on X in the usual sense.

The set of partial functions from X to Y is partial order by refinement: we say that f refines g if the domain of definition of f contains that of g and f agrees with g on the domain of g.