Signum Function is Primitive Recursive

Theorem
Let $\operatorname{sgn}: \N \to \N$ be defined as the signum function.

Then:
 * $\operatorname{sgn}$ is primitive recursive.
 * $\overline {\operatorname{sgn}}$ is primitive recursive.

Proof
We have that the characteristic function $\chi_{\N^*}$ of $\N^*$, where $\N^* = \N \setminus \left\{{0}\right\}$, is primitive recursive.

We also have by definition that $\operatorname{sgn} \left({n}\right) = \chi_{\N^*} \left({n}\right)$.

Thus $\operatorname{sgn}$ is primitive recursive.

Now $\N - \N^* = \left\{{0}\right\}$ from Relative Complement of Relative Complement.

We also have by definition that $\overline {\operatorname{sgn}} \left({n}\right) = \chi_{\left\{{0}\right\}} \left({n}\right)$.

Thus $\overline {\operatorname{sgn}}$ is primitive recursive from Complement of Primitive Recursive Set.