Mathwords: Closed Interval

Q: What is the difference between a closed interval and an open interval?

A closed interval $[a, b]$ includes both endpoints, so $a$ and $b$ are part of the set. An open interval $(a, b)$ excludes both endpoints, meaning $a$ and $b$ are not part of the set. Graphically, closed intervals use filled-in dots at the endpoints, while open intervals use hollow circles.

Q: When do you use square brackets versus parentheses in interval notation?

Use square brackets $[\;]$ when the endpoint is included in the interval (closed side) and parentheses $(\;)$ when the endpoint is excluded (open side). For example, $[2, 5)$ includes 2 but not 5. You always use a parenthesis next to $\infty$ or $-\infty$ because infinity is not a real number and cannot be reached.

Q: Can a closed interval contain just one point?

Yes. If $a = b$, then $[a, a] = \{a\}$, which is a valid closed interval containing exactly one point. This is sometimes called a degenerate interval. In contrast, the open interval $(a, a)$ would be empty.

Key Formula

[a,\, b] = \{\, x \in \mathbb{R} \;|\; a \le x \le b \,\}

Where:

$a$ = The lower endpoint (left boundary) of the interval
$b$ = The upper endpoint (right boundary) of the interval
$x$ = Any real number that satisfies $a \le x \le b$

Worked Example

Problem: Determine whether the values 1, 3.5, and 7 belong to the closed interval

[1, 7]

.

Step 1: Write out the condition for membership in the interval. A number

x

is in

[1, 7]

when:

1 \le x \le 7

Step 2: Test

x = 1

. Since

1 \le 1 \le 7

is true, the value 1 is in the interval. Notice that the endpoint itself is included because the interval is closed.

1 \le 1 \le 7 \quad \checkmark

Step 3: Test

x = 3.5

. Since

1 \le 3.5 \le 7

is true, this value is also in the interval.

1 \le 3.5 \le 7 \quad \checkmark

Step 4: Test

x = 7

. Since

1 \le 7 \le 7

is true, the upper endpoint 7 is included as well.

1 \le 7 \le 7 \quad \checkmark

Answer: All three values — 1, 3.5, and 7 — belong to the closed interval

[1, 7]

.

Another Example

This example differs by using negative numbers and by computing the interval's length, showing that a closed interval contains infinitely many real numbers even though only six of them are integers.

Problem: List all integers in the closed interval

[-2, 3]

and find the length of the interval.

Step 1: Identify the membership condition:

x

is in

[-2, 3]

when

-2 \le x \le 3

.

-2 \le x \le 3

Step 2: List the integers satisfying this inequality. Because the interval is closed,

-2

and

3

are both included.

\{-2,\, -1,\, 0,\, 1,\, 2,\, 3\}

Step 3: Calculate the length of the interval by subtracting the lower endpoint from the upper endpoint.

\text{Length} = 3 - (-2) = 5

Answer: The integers in

[-2, 3]

are

\{-2, -1, 0, 1, 2, 3\}

, and the interval has length 5.

Frequently Asked Questions

What is the difference between a closed interval and an open interval?

A closed interval

[a, b]

includes both endpoints, so

a

and

b

are part of the set. An open interval

(a, b)

excludes both endpoints, meaning

a

and

b

are not part of the set. Graphically, closed intervals use filled-in dots at the endpoints, while open intervals use hollow circles.

When do you use square brackets versus parentheses in interval notation?

Use square brackets

[\;]

when the endpoint is included in the interval (closed side) and parentheses

(\;)

when the endpoint is excluded (open side). For example,

[2, 5)

includes 2 but not 5. You always use a parenthesis next to

\infty

or

-\infty

because infinity is not a real number and cannot be reached.

Can a closed interval contain just one point?

Yes. If

a = b

, then

[a, a] = \{a\}

, which is a valid closed interval containing exactly one point. This is sometimes called a degenerate interval. In contrast, the open interval

(a, a)

would be empty.

Closed Interval vs. Open Interval

	Closed Interval	Open Interval
Notation	$[a, b]$ — square brackets	$(a, b)$ — parentheses
Endpoints	Both $a$ and $b$ are included	Neither $a$ nor $b$ is included
Inequality	$a \le x \le b$	$a < x < b$
Number line graph	Filled (solid) dots at endpoints	Hollow (open) circles at endpoints
Contains its boundary?	Yes — the interval equals its closure	No — the endpoints are outside the set
Example: does 5 belong?	$5 \in [3, 5]$ — yes	$5 \notin (3, 5)$ — no

Why It Matters

Closed intervals appear throughout algebra and calculus. When you state the domain or range of a function, choosing a closed versus open interval determines whether boundary values are valid. In calculus, the Extreme Value Theorem guarantees that a continuous function on a closed interval

[a, b]

attains both a maximum and a minimum — a result that fails on open intervals, making the distinction critical for optimization problems.

Common Mistakes

Mistake: Using parentheses when you mean to include the endpoints, writing

(2, 5)

instead of

[2, 5]

.

Correction: Remember: square brackets mean "included" (think of them as walls that keep the endpoint in). Parentheses mean "excluded." Double-check your inequality:

\le

pairs with

[\;]

, while

<

pairs with

(\;)

.

Mistake: Writing

[3, \infty]

with a square bracket next to infinity.

Correction: Infinity is not a real number, so it can never be "reached" or included. Always write

[3, \infty)

with a parenthesis on the infinity side.

Related Terms

Interval — General concept that closed intervals are a type of
Interval Notation — Notation system using brackets and parentheses
Open Interval — Interval that excludes both endpoints
Half-Open/Half-Closed Interval — Interval that includes one endpoint but not the other
Domain — Often expressed using closed intervals
Range — Output set frequently written as a closed interval
Inequality — Closed intervals correspond to ≤ inequalities