Value of Multiplicative Function at One

Theorem
Let $f: \N \to \C$ be a multiplicative function.

If $f$ is not identically zero, then $f \left({1}\right) = 1$.

Proof
If $f$ is not identically zero, then:


 * $\exists m \in \Z: f \left({m}\right) \ne 0$

Then:


 * $f \left({m}\right) = f \left({1 \times m}\right) = f \left({1}\right) f \left({m}\right)$

Hence $f \left({1}\right) = 1$.