General Periodicity Property/Corollary

Theorem
Let $f: \R \to \R$ be a real function.

Then $f$ is periodic with period $L$ if and only if:
 * $f \left({x \bmod L}\right) = f \left({x}\right)$

for all $x \in \R$, where $x \bmod L$ is the modulo operation.

Proof
Let $f: \R \to \R$ be a real periodic function with period $L$, then:

For the converse, let $f: \R \to \R$ be a real function such that for all $x \in \R$, $f \left({x \bmod L}\right) = f \left({x}\right)$.

Let $n = \left \lfloor {\dfrac x L}\right \rfloor$.

Then: