Definition:Periodic Function

Real Function
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 $$L \in \mathbb{R}$$ such that $$f \left({x}\right) = f \left({x + L}\right)$$ for all $$x \in \R$$.