Periodicity of Real Functions

Periodicity

Definition: Periodic Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is periodic iff there exists a $P\ne 0$ such that

$$f(x+P)=f(x)\forall x\in D$$

Definition: Antiperiodic Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is antiperiodic iff there exists a $P\ne 0$ such that

$$f(x+P)=-f(x)\forall x\in D$$

Theorem: Modifications of Periodic Functions

If $f(x)$ has a period of $P$, then $f(\alpha x)$, where $\alpha \in \mathbb{R}$ has a period of $\frac{P}{\alpha }$, provided that $\alpha \ne 0$.

PROOF

TODO