Signum Function is Primitive Recursive

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

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

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

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

Thus $$\sgn$$ is primitive recursive.

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

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

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