Periodicity · Little Blue Book, Visualized

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Choose “triangle wave”. For which values of $T$ does the largest mismatch drop to $0$ ? How are those values related?
Choose “fractional part ${x}$ ”, then “triangle + $\frac{1}{2} {x /3}$ ”. What are their least positive periods? How is the last one related to $2$ and $3$ ?
Choose “symmetric humps” and switch on “Two mirrors”. The point $x_{0}$ is reflected in $x = 0$ and then in $x = 1$ . How far has it moved in total?
If $T$ is a period, are $2 T$ and $3 T$ periods too? Check on the slider.

If there is a non-zero constant $T$ such that, for every $x$ in the domain of $f (x)$ ,

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

then $f (x)$ is a periodic function and $T$ is one of its periods. The smallest of all positive periods (if there is one) is the least positive period. In the picture: shift the whole graph right by $T$ and it lands exactly on itself.

Predict first

$f$ has period $2$ and $g$ has period $3$ , both defined on $R$ . Which of these is certainly a period of $f + g$ ?

2

3

5

6

Choose an answer first; then you can see the result.

Why: two reflections make one translation

Example: f is even, and its graph is also symmetric about the line x = 1. Prove that 2 is a period of f.

Step 1 / 5

Two mirrors
“Even” says the graph is symmetric about $x = 0$ ; “symmetric about $x = 1$ ” says $f (1 + u) = f (1 - u)$ for every $u$ . The two dash-dotted lines in the figure are these two mirrors.

$f (- x) = f (x), f (1 + u) = f (1 - u)$
First reflection
Second reflection
A total move of 2
The general fact

Some periods can be spotted without any picture. They all rely on “do it twice (or several times) and you are back where you started”:

Some common “hidden periods”

Step 1 / 3

f(x+a) = −f(x)
Do it twice: $f (x + 2 a) = - f (x + a) = f (x)$ , so $2 a$ is a period.

$f (x + 2 a) = f (x)$
f(x+a) = 1/f(x)
f(x+a) = g(f(x))

Practice

Exercise 1Basic

$f (x)$ has period $3$ and $f (- 1) = 7$ . Find $f (2027)$ .

Exercise 2Basic

$f (x)$ is defined on $R$ , $f (x + 2) = - f (x)$ and $f (1) = 2$ . Find $f (7)$ .

Exercise 3Advanced

$f (x)$ is an even function on $R$ whose graph is symmetric about the line $x = 2$ , and $f (x) = x^{2}$ for $0 \leq x \leq 2$ . Find $f (7)$ .

Exercise 4Challenge

$f (x)$ is defined on $R$ , never takes the value $- 1$ , and satisfies $f (x + 3) = \frac{f ( x ) - 1}{f ( x ) + 1}$ and $f (1) = 2$ . Find $f (2026)$ .

Mistake Museum · Exhibits from this stop

Before you leave, look at the traps other people fell into here.See every exhibit in the museum →

Exhibit

Seeing $f(x+2)=-f(x)$ and calling $2$ the period

f (x + 2) = - f (x)

, so

f (x)

is periodic with period

2

What went wrong? Think first, then open

Diagnosis After a shift by $2$ the graph does not land on itself: it lands upside down. A period needs $f (x + T) = f (x)$ , and here there is an extra minus sign.

Shift once more:

f (x + 4) = - f (x + 2) = - (- f (x)) = f (x)

. The period is

4

(and

2

is not a period unless

f

is identically

0

“Flip twice to get back to where you started”

Exhibit

Every periodic function has a least positive period

Every periodic function has a least positive period; just work it out.

What went wrong? Think first, then open

Diagnosis The constant function $f (x) = c$ has every positive number as a period, so there is no smallest one. The Dirichlet function ( $1$ when $x$ is rational, $0$ when $x$ is irrational) has every positive rational as a period, and also has no least positive period.

Before you talk about “the least positive period”, make sure it exists. When we say “

sin x

has period

2 π

”, we mean its least positive period.

“Having a period does not mean having a least positive period”