Interval

An interval is a connected subset of the real line. Intervals are the standard way to describe continuous domains for real-valued functions.

Endpoint notation

For real numbers $a<b$:

• $(a,b)=\{x\in \mathbb{R}:a<x<b\}$ excludes both endpoints.
• $[a,b]=\{x\in \mathbb{R}:a\le x\le b\}$ includes both endpoints.
• $[a,b)$ includes $a$ but excludes $b$.
• $(a,b]$ excludes $a$ but includes $b$.
• $(a,\infty )=\{x\in \mathbb{R}:x>a\}$ is unbounded above.

Infinity is never included as an endpoint: write $(a,\infty )$, not $(a,\infty ]$.

Why intervals matter

Many calculus theorems require functions on intervals, often with extra conditions such as continuity on $[a,b]$ or differentiability on $(a,b)$. Closed intervals are especially important because continuous real-valued functions on them attain maximum and minimum values.

Applied examples

Time in a simple experiment might be $t\in [0,T]$. A radial coordinate might satisfy $r\in [0,\infty )$. Probabilities often integrate over intervals of possible values. In mechanics, a trajectory can be written as a mapping from a time interval into space, e.g. $x:[0,T]\to {\mathbb{R}}^3$.

Related: Number sets, Set, Derivative, Integral.

Why intervals matter

Intervals are the natural domains for one-dimensional calculus. For example, a derivative may exist on an open interval

$$(a,b)$$

while endpoint behaviour needs one-sided limits. In optimisation, closed intervals like

$$[a,b]$$

are special because continuous functions on them attain maxima and minima. Tiny bracket choice, massive theorem consequences.