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.