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".

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