General Periodicity Property/Corollary

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

Let $L$ be the period of $f$.

Then $f$ can be defined entirely on the closed interval $\left[{0 \,.\,.\, \left\lvert{L}\right\rvert}\right]$, and
 * $f \left({x \bmod L}\right) = f \left({x}\right)$

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

Proof
It is seen that for all $x \in \R$:

This implies that
 * $f: \R \to \R$

can be determined entirely by the restriction:
 * $f: \left[{0 \,.\,.\, \left\lvert{L}\right\rvert}\right] \to \R$

Hence the result.