Definition:Periodic Function

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

Then $f$ is referred to as periodic iff:
 * $\exists L \in \R: \forall x \in \R: f \left({x}\right) = f \left({x + L}\right)$

It follows immediately that if $f$ is periodic, then:
 * $\forall n \in \Z: \forall x \in \R: f \left({x}\right) = f \left({x + nL}\right)$

That is, after every distance $L$, the function $f$ repeats itself.

Period
The period of $f$ is the smallest $L \in \R$ such that $f \left({x}\right) = f \left({x + L}\right)$ for all $x \in \R$.