General Periodicity Property/Corollary

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

Then $L$ is a periodic element of $f$ :
 * $\forall x \in \R: \map f {x \bmod L} = \map f x$

where $x \bmod L$ is the modulo operation.

Necessary Condition
Let $f: \R \to \R$ be a real function with a periodic element $L$.

Then:

Sufficient Condition
Let $f: \R \to \R$ be a real function such that for all $x \in \R$:
 * $\map f {x \bmod L} = \map f x$

Let $n = \floor {\dfrac x L}$.

Then: