Signum Function is Completely Multiplicative

Theorem
The signum function on the set of real numbers is a completely multiplicative function:


 * $\forall x, y \in \R: \map \sgn {x y} = \map \sgn x \, \map \sgn y$

Proof
Let $x = 0$ or $y = 0$.

Then:

and either $\map \sgn x = 0$ or $\map \sgn y = 0$ and so:

Let $x > 0$ and $y > 0$.

Then:

and:

Let $x < 0$ and $y < 0$.

Then:

and:

Let $x < 0$ and $y > 0$.

Then:

and:

The same argument, mutatis mutandis, covers the case where $x > 0$ and $y < 0$.