Interval¶

Definition¶

Let $a,b\in \mathbb{R}$ be real numbers with $a\leq b$ $b$ .

We define the closed intervals $[a, b]$ and the open intervals $(a,b)$ as follows:

\begin{aligned} &[a,b] = \{x\in \mathbb{R}\colon a\leq x\leq b\},\\ &(a,b) = \{x\in \mathbb{R}\colon a< x< b\}. \end{aligned}

In the case when $a<b\text{,}$ we define the half-open intervals $[a, b)$ and $(a, b]$ as follows:

\begin{aligned} &[a,b) = \{x\in \mathbb{R}\colon a\leq x< b\},\\ & (a,b] = \{x\in \mathbb{R}\colon a< x\leq b\}. \end{aligned}

We define open half-infinite interval as follows:

\begin{aligned} &(-\infty,b)=\{x\in \mathbb{R}\colon x< b\},\\ & (a,+\infty) = \{x\in \mathbb{R}\colon x>a\}. \end{aligned}

We define closed half-infinite interval as follows:

\begin{aligned} &(-\infty,b]=\{x\in \mathbb{R}\colon x\leq b\},\\ & [a,+\infty) = \{x\in \mathbb{R}\colon x\geq a\}. \end{aligned}

Interpretation¶

Intervals are exactly the subsets of the real line without gaps: whenever two numbers lie in an interval, every number between them does too. They are the usual domains of the functions studied in this course.

Number lines showing the intervals from a to b: closed with both endpoints filled, open with both endpoints hollow, and the two half-open intervals with one endpoint filled.

The four number lines show the intervals from $a$ to $b$ . A filled dot marks an endpoint that belongs to the interval and a hollow dot one that does not: $[a,b]$ has both endpoints, $(a,b)$ neither, and each half-open interval has one. In the notation, a square bracket corresponds to a filled dot and a round bracket to a hollow one.

Each interval is a subset of the real numbers defined by inequalities. Weak inequalities $\le$ include the endpoint and strict inequalities $<$ exclude it. The symbols $-\infty$ and $+\infty$ are not real numbers, so they always take a round bracket: one writes $[a,+\infty)$ , never $[a,+\infty]$ . The half-infinite intervals, together with $\mathbb{R}=(-\infty,+\infty)$ itself, are the unbounded intervals; the others are bounded. The case $a=b$ is allowed for the closed and open intervals: $[a,a]=\{a\}$ is a singleton and $(a,a)$ is the empty set.

Whether the endpoints belong to the interval often matters: $[0,1]$ has maximum $1$ , while $(0,1)$ has no maximum, although its supremum is still $1$ . A common slip is to read $(a,b)$ as a point of the plane. The same symbols denote an ordered pair of coordinates, and only the context tells which is meant.

Key Results¶

Intervals may be open, closed, or half-open.
Endpoints determine whether boundary values are included.

Examples¶

$(0,1)$ is an open interval.
$[0,1]$ includes both endpoints.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Symbols: element of, infinity, real numbers

Last updated: 20 October 2026 03:02