Signum Complement Function is Primitive Recursive

Theorem
Let $\overline {\operatorname{sgn}}: \N \to \N$ by defined as the signum-bar function.

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

Proof
From Signum Complement Function on Natural Numbers as Characteristic Function, $\map {\overline {\operatorname{sgn} } } n = \chi_{\set 0} n$.

From Set Containing Only Zero is Primitive Recursive, $\chi_{\set 0}$ is primitive recursive.

Hence the result.