# Periodic Function

Function $$$y={f{{\left({x}\right)}}}$$$ is called **periodic** if exists such number $$${T}\ne{0}$$$ that for any $$${x}$$$ from domain of the function $$${f{{\left({x}+{T}\right)}}}={f{{\left({x}\right)}}}$$$.

From the definition it follows that periodic function has infinitely many periods. If $$$T$$$ is a period of a function then any number of the form $$${k}{T}$$$, where $$$k$$$ is an integer,is also a period of the function.

Often (but not always) among set of positive periods of the function we can find the smallest one. This period is called **main period** (or simply **period).**

For example trigonometric function $$${y}={\sin{{\left({x}\right)}}}$$$ has period $$${2}\pi$$$ because $$${\sin{{\left({x}+{2}\pi\right)}}}={\sin{{\left({x}\right)}}}$$$ for all $$${x}$$$.