Definition:Periodic Function

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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.

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