We want to discuss a function which repeats itself on regular intervals.1
A function $f: \R \to \R $ is periodic with period $\alpha \in \R $ if \[ f(x + \alpha ) = f(x) \] for all $x \in \R $. In this case we call $f$ $\alpha $-periodic.
\[ f(x + \alpha ) = f(x) \]