Value of Multiplicative Function at One

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

If $f$ is not identically zero, then $\map f 1 = 1$.

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


 * $\exists m \in \Z: \map f m \ne 0$

Then:


 * $\map f m = \map f {1 \times m} = \map f 1 \, \map f m$

Hence $\map f 1 = 1$.