Introduction to Periodic Functions

A periodic function repeats its values at regular intervals. Periodicity appears throughout mathematics and the sciences: in the oscillation of springs, the motion of tides, the rotation of planets, and alternating electrical current.

A schematic graph of a periodic function on the xy-plane. One full cycle is drawn in solid blue, showing one complete wave with a crest above and a trough below the x-axis. The remaining cycles extend in both directions as dashed blue curves, indicating the pattern repeats indefinitely. A red double-headed arrow below the solid cycle is labeled T, marking the length of one period. — A periodic function repeats the same pattern every T units along the x-axis, where T is the period.

Quick Reference

Concept	Meaning
Periodic with period	for all in the domain
Period not unique	If is a period, so are , , , etc.
Fundamental period	The smallest positive period of (if it exists)
Constant functions	Periodic with every ; no fundamental period

Definition

A function is periodic with period if:

Whenever lies in the domain of , so does , and
for every in the domain of .

The first condition ensures the domain is itself "periodic," allowing the shift to make sense. A function need not be defined for all real numbers to be periodic.

Multiple Periods and the Fundamental Period

The period of a periodic function is not unique. If is a period of , then , , , etc., are also periods:

In general:

If is periodic with period , then for every integer .

The smallest positive period of a periodic function (if it exists) is called the fundamental period of the function.

A constant function is periodic with every positive as a period, since . Because there is no smallest positive number, a constant function has no fundamental period.

Examples

Find the fundamental period of .

Solution

Suppose

is a period of

. Then

for all

Expanding:

Canceling

from both sides:

The right side is always an integer (the difference of two integers), so

must be an integer. To find the fundamental period, we choose

to be the smallest positive integer, which is

The fundamental period of

Graph of the periodic function f(x) = 2x - ⌊2x⌋ showing a sawtooth pattern. — Graph of : a sawtooth wave with fundamental period .

Show that is not periodic.

Solution

Assume for contradiction that

has period

. Then

for all

, which means:

Expanding

Canceling

The right side is always an integer (difference of two floor values). So the left side

must be an integer for every real number

. But

is a linear function of

with slope

. If

, this linear function takes all real values as

varies, not just integer values. This is a contradiction. Therefore no such

exists, and

is not periodic.

Graph of f(x) = x² - ⌊x²⌋ showing a non-periodic, irregular pattern. — Graph of : despite appearing to have a pattern, this function is not periodic.

Frequently Asked Questions

Can a function have more than one period?

Yes. If

is a period, then

, and more generally

for any nonzero integer

are also periods. The set of all periods is closed under integer multiples. The fundamental period is the smallest positive one, if it exists.

Does every periodic function have a fundamental period?

No. A constant function

satisfies

for every positive

. Since there is no smallest positive real number, a constant function has no fundamental period. More exotic examples also exist, but they are not encountered in precalculus.

How can I check if a given function is periodic?

One approach is to assume a period

exists and derive what conditions

must satisfy from the equation

. If those conditions lead to a valid positive

, the function is periodic. If the conditions lead to a contradiction (as in the

example), the function is not periodic.

What is the difference between period and fundamental period?

Any positive number

satisfying

for all

is called a period of

. A function typically has infinitely many periods. The fundamental period is the smallest among all positive periods. For example,

has periods

, and its fundamental period is

Periodic Functions

Quick Reference

Definition

Multiple Periods and the Fundamental Period

Examples

Frequently Asked Questions

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