# 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 $L \in \R_{>0}$ such that:

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

### Frequency

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

The frequency $\nu$ of $f$ is the reciprocal of the period $L$ of $f$:

$\nu = \dfrac 1 L$

### Amplitude

The amplitude of $f$ is the maximum absolute difference of the value of $f$ from a reference level.

## 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.

## Examples

### Sawtooth Function

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x - \floor x$

where $\floor x$ denotes the floor function.

$f$ is periodic with period $1$.

## Also see

• Results about periodic functions can be found here.