# Definition:Periodic Function/Real

## Definition

Let $f: \R \to \R$ be a real function.

Then $f$ is periodic if and only if:

$\exists L \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + L}$

### Period

The period of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that:

$\forall x \in X: \map f x = \map f {x + L}$

where $\cmod L$ is the modulus of $L$.

## Also defined as

Some sources define a periodic real function as one such that:

$\exists L \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + 2 L}$

In some circumstances it can make the algebra simpler to define the period as being $2$ times a specified constant.

## Also see

• Results about periodic functions can be found here.