linguistic:paraphrasing, stylistic:tone
Real number examples

The sine function is periodic with period 2 π {\displaystyle 2\pi }, since

for all values of x {\displaystyle x}. This function repeats on intervals of length 2 π {\displaystyle 2\pi } (see the graph to the right).

Everyday examples are seen when the variable is time; for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.

For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function f {\displaystyle f} that gives the " fractional part " of its argument. Its period is 1. In particular,

The graph of the function f {\displaystyle f} is the sawtooth wave.

The trigonometric functions sine and cosine are common periodic functions, with period 2 π {\displaystyle 2\pi } (see the figure on the right). The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.