Signum Function on Natural Numbers as Characteristic Function

Definition
Let $\sgn: \N \to \N$ be the signum function on the natural numbers.

Let $\chi_{\N_{>0} }: \N \to \set {0, 1}$ be the characteristic function of $\N_{>0}$, where $\N_{>0} = \N \setminus \set 0$.

Let $n \in \N$.

Then:
 * $\sgn \paren n = \chi_{\N_{>0} } \paren n$

Proof
The signum function $\sgn: \N \to \N$ is defined as:
 * $\forall n \in \N: \sgn \paren n = \begin{cases}

0 & : n = 0 \\ 1 & : n > 0 \end{cases}$

The result follows by definition of the characteristic function.