Textbook
For real numbers $a<b$, the closed interval $[a,b]=\{x\in\mathbb R\mid a\le x\le b\}$ includes both endpoints. The open interval $(a,b)=\{x\in\mathbb R\mid a<x<b\}$ excludes both. A square bracket includes its endpoint; a round bracket excludes it. Thus $[a,b)$ includes only the left endpoint, and $(a,b]$ only the right one.
An interval contains real values between the endpoints, not just integers. For example, $[2,4)\cap\mathbb Z=\{2,3\}$, while the interval itself also contains 2.5. Infinity is not a real endpoint that can be included: the notation is $[0,+\infty)$. In another context, $(2,4)$ may instead denote an ordered pair.
Membership in $[a,b)$ requires both comparisons, $a\le x$ and $x<b$. Satisfying just one is insufficient. Each bracket governs its own endpoint and leaves interior points unchanged. If equal endpoints are allowed, $[a,a]=\{a\}$, while $(a,a)$, $[a,a)$, and $(a,a]$ are empty: a strict bound rejects the only possible candidate. The initial assumption $a<b$ therefore specifies the nondegenerate case.
Definitions and further reading: Basic Analysis I.
In plain language
An interval resembles a stretch of road with separate admission rules at its endpoints. Every real coordinate inside is included, not just the integer markers. The only difference between $[2,4)$ and $[2,4]$ is the point 4, but a deadline boundary can give one point enough influence to convene a meeting. Coincident endpoints leave either one admitted point or nothing; rounded measurements do not change these rules.
Example
The example tests membership in the illustrative half-open interval $[2,4)$. The left endpoint and interior values are included; the right endpoint is excluded.
Real Intervals: Are the Endpoints Included?
Left endpoint
The square bracket admits 2.
Step 1 of 3: Left endpoint
Read all steps
Left endpoint
The square bracket admits 2.
- State
- 2included2.53.994
Interior points
Fractional values between the endpoints also belong.
- State
- 2included2.5included3.99included4
Right endpoint
The round bracket excludes 4.
- State
- 2included2.5included3.99included4excluded
For 3.99, both comparisons succeed: it is at least 2 and strictly below 4. At four itself, the second comparison fails despite the small difference from the previous candidate. There is no “last permitted number” before four: another interior number lies between any chosen interior point and that endpoint. The viewer shows individual checks; the inequalities specify the entire interval without enumerating its points.
Related concept: Set-builder notation.