Periodic function

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 with period $2 \pi$. If a continuous periodic function is not constant, it has a smallest positive period. The Dirichlet function 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.