Definition:Partial Function

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

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

Suppose that $\forall x \in \N^k \setminus S$, $f$ 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 $\N$.

It can be seen that the definition of a partial function as given here is compatible with that of a partial mapping.

Total Function
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.