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 $T$	$f (x + T) = f (x)$ for all $x$ in the domain
Period not unique	If $T$ is a period, so are $2 T$ , $3 T$ , $- T$ , etc.
Fundamental period	The smallest positive period of $f$ (if it exists)
Constant functions	Periodic with every $T$ ; no fundamental period

Definition

A function $f$ is periodic with period $T$ if:

Whenever $x$ lies in the domain of $f$ , so does $x + T$ , and
$f (x + T) = f (x)$ for every $x$ in the domain of $f$ .

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 $T$ is a period of $f$ , then $2 T$ , $3 T$ , $- T$ , etc., are also periods:

$f(x + 2T) = f((x+T) + T) = f(x+T) = f(x),$

f (x - T) = f (x - T + T) = f (x) .

In general:

If $f$ is periodic with period $T$ , then $f (x + n T) = f (x)$ for every integer $n$ .

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

A constant function $f (x) = c$ is periodic with every positive $T$ as a period, since $f (x + T) = c = f (x)$ . Because there is no smallest positive number, a constant function has no fundamental period.

Examples

Find the fundamental period of $f (x) = 2 x - ⌊ 2 x ⌋$ .

Solution

Suppose

T

is a period of

f

. Then

f (x + T) = f (x)

for all

x

2 (x + T) - ⌊ 2 (x + T) ⌋ = 2 x - ⌊ 2 x ⌋ .

Expanding:

2 x + 2 T - ⌊ 2 (x + T) ⌋ = 2 x - ⌊ 2 x ⌋ .

Canceling

2 x

from both sides:

2 T = ⌊ 2 (x + T) ⌋ - ⌊ 2 x ⌋ .

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

2 T

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

2 T

to be the smallest positive integer, which is $1$:

2 T = 1 ⟹ T = \frac{1}{2} .

The fundamental period of

f

\frac{1}{2}

Graph of the periodic function f(x) = 2x - ⌊2x⌋ showing a sawtooth pattern. — Graph of $f (x) = 2 x - ⌊ 2 x ⌋$ : a sawtooth wave with fundamental period $T = 1 / 2$ .

Show that $f (x) = x^{2} - ⌊ x^{2} ⌋$ is not periodic.

Solution

Assume for contradiction that

f

has period

T

. Then

f (x + T) = f (x)

for all

x

, which means:

(x + T)^{2} - ⌊ (x + T)^{2} ⌋ = x^{2} - ⌊ x^{2} ⌋ .

Expanding

(x + T)^{2} = x^{2} + 2 x T + T^{2}

x^{2} + 2 x T + T^{2} - ⌊ (x + T)^{2} ⌋ = x^{2} - ⌊ x^{2} ⌋ .

Canceling

x^{2}

2 x T + T^{2} = ⌊ (x + T)^{2} ⌋ - ⌊ x^{2} ⌋ .

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

2 x T + T^{2}

must be an integer for every real number

x

. But

2 x T + T^{2}

is a linear function of

x

with slope

2 T

. If

T \neq 0

, this linear function takes all real values as

x

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

T

exists, and

f

is not periodic.

Graph of f(x) = x² - ⌊x²⌋ showing a non-periodic, irregular pattern. — Graph of $f (x) = x^{2} - ⌊ x^{2} ⌋$ : despite appearing to have a pattern, this function is not periodic.

Frequently Asked Questions

Can a function have more than one period?

Yes. If

T

is a period, then

2 T

3 T

- T

, and more generally

n T

for any nonzero integer

n

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

f (x) = c

satisfies

f (x + T) = c = f (x)

for every positive

T

. 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

T

exists and derive what conditions

T

must satisfy from the equation

f (x + T) = f (x)

. If those conditions lead to a valid positive

T

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

x^{2} - ⌊ x^{2} ⌋

example), the function is not periodic.

What is the difference between period and fundamental period?

Any positive number

T

satisfying

f (x + T) = f (x)

for all

x

is called a period of

f

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

\sin (x)

has periods

2 π, 4 π, 6 π, \dots

, and its fundamental period is

2 π

Periodic Functions

Quick Reference

Definition

Multiple Periods and the Fundamental Period

Examples

Frequently Asked Questions

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