A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves and other repeating phenomena, are periodic. Many aspects of the natural world have periodic behavior, such as the phases of the Moon, the swinging of a pendulum, and the beating of a heart.

The length of the interval over which a periodic function repeats is called its period. Any function that is not periodic is called aperiodic.

A function is defined as periodic if its values repeat at regular intervals. For example, the positions of the hands on a clock display periodic behavior as they cycle through the same positions every 12 hours. This repeating interval is known as the period.

More formally, a function ${\displaystyle f}$ is periodic if there exists a constant ${\displaystyle P}$ such that

for all values of ${\displaystyle x}$ in the domain. A nonzero constant ${\displaystyle P}$ for which this condition holds is called a period of the function.

If a period ${\displaystyle P}$ exists, any integer multiple ${\displaystyle nP}$ (for a positive integer ${\displaystyle n}$) is also a period. If there is a least positive period, it is called the fundamental period (also primitive period or basic period). Often, "the" period of a function is used to refer to its fundamental period.

Geometrically, a periodic function's graph exhibits translational symmetry. Its graph is invariant under translation in the ${\displaystyle x}$-direction by a distance of ${\displaystyle P}$. This implies that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

Periodic behavior can be illustrated through both common, everyday examples and more formal mathematical functions.