Endpoint and Bounded Interval¶

Definition¶

Let $a,b\in \mathbb{R}$ be real numbers such that $a<b\text{.}$ The numbers $a$ and $b$ are called the endpoints of the intervals (either $(a,b)\text{,}$ $[a,b]\text{,}$ $(a,b]$ or $[a,b)$), and any number $x$ with $a<x<b$ is called an interior point of the interval. An interval contained in $[u,v]$ for some real $u\le v$ is called a bounded interval.

The closed interval $[a,a]=\{a\}$ is a singleton whose endpoints coincide. The empty interval $(a,a)=\varnothing$ is also bounded as a set, but it has no intrinsic endpoints: the same empty set can be written $(c,c)$ for any real $c$.

The real line itself has no endpoints. The symbols $+\infty$ and $-\infty$ describe unbounded directions; they are not real numbers and are never elements of an interval.

Interpretation¶

The endpoints of an interval are the real numbers where it starts and stops. They fix the extent of the interval; whether they belong to it is a separate choice, recorded by the type of bracket.

The figure shows the closed interval $[a,b]$, which contains both endpoints. The intervals $(a,b)$, $(a,b]$ and $[a,b)$ cover the same stretch of the line and differ only at $a$ and $b$; all four have the same interior points $a<x<b$ and the same width $b-a$$a$.

A nonempty bounded interval has finite real endpoints, which may coincide for a singleton. A ray such as $[a,+\infty)$ has only one, and $\mathbb{R}$ has none, because $\pm\infty$ are not real numbers.

A common slip is to assume that an endpoint belongs to the interval: $0$ is an endpoint of $(0,1]$, but $0\notin(0,1]$$(0,1]$. In Math III, the endpoints return as the boundary of the interval.

Key Results¶

• An interval with two endpoints is bounded.
• A bounded interval is a specific type of bounded set in $\mathbb{R}\text{.}$
• Bounded intervals have finite length.

Examples¶

• $[0,1]$ has endpoints $0$ and $1\text{.}$
• $(0,1]$ is bounded but does not include its left endpoint.

Prerequisites¶

• Interval

Related Concepts¶

• Boundary of a Set

Sources¶

• Math I: Calculus · Lecture 4

Aliases: bounded interval Symbols: element of, infinity, real numbers

Last updated: 27 October 2026 11:52