Completely Multiplicative Function of Quotient

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Theorem

Let $f: \R \to \R$ be a completely multiplicative function.

Then:

$\forall x, y \in \R, y \ne 0: \map f {\dfrac x y} = \dfrac {\map f x} {\map f y}$

whenever $\map f y \ne 0$.


Proof

Let $z = \dfrac x y$.

Then:

\(\ds z\) \(=\) \(\ds \dfrac x y\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds y z\)
\(\ds \leadsto \ \ \) \(\ds \map f x\) \(=\) \(\ds \map f y \map f z\) Definition of Completely Multiplicative Function
\(\ds \leadsto \ \ \) \(\ds \frac {\map f x} {\map f y}\) \(=\) \(\ds \map f z\) dividing by $\map f y$
\(\ds \leadsto \ \ \) \(\ds \frac {\map f x} {\map f y}\) \(=\) \(\ds \map f {\frac x y}\) Definition of $z$

$\blacksquare$