Definition:Partial Function

Let $$S \subset \N^k$$.

Let $$f: S \to \N$$ be a function.

Suppose that $$\forall x \in \N^k - S$$, $$f$$ is undefined at $$x$$.

Then $$f$$ is known as a partial function from $$\N^k$$ to $$\N$$.

Thus we can specify a function that has values for some, but not all, elements of $$\N$$.

Total
If it is necessary to emphasise the fact that the domain of $$f$$ is indeed the whole of $$\N^k$$, then in that case we can say that $$f$$ is a total function.