Periodicity of Real Functions

Definition: Periodic Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is period 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

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