---
tag: 089KWVYD69
kind: definition
name: Periodic function
msc: [26A09]
defines: 0ATF6Q0633
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_function
    locator: Definition
---

## Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function and let $T$ be a real number.
The number $T$ is a period of $f$ if $f(x + T) = f(x)$ for all $x \in \mathbb{R}$.
The function $f$ is periodic if it has a period $T \ne 0$.

## Remarks

If $T$ is a period, then so is $k T$ for each integer $k$.
The [[sine function is periodic|0P4NQD746K]] with period $2 \pi$.
If a continuous periodic function is not constant, it has a smallest positive period. The [[Dirichlet function|0T4EJ8ZJM9]] has every rational number as a period.
Mathlib defines periods as `Function.Periodic`.

## Explanations

### L1 human

A periodic function repeats itself after each step of length $T$.
Its graph is made of copies of one piece, placed side by side.