A bracket or a parenthesis decides whether a number belongs to an interval
Write the numbers from 0 to 1 with square brackets and both ends are included; swap one bracket for a round parenthesis and that endpoint vanishes from the set. This tiny notational choice sits underneath continuity, integration and guaranteed-accurate computing, all of which lean on the humble mathematical interval.
An interval is a stretch of real numbers with no gaps: whenever two numbers belong to it, so does everything between them. The set from 0 to 1 including both ends is called the unit interval. Leave out both endpoints and it is open; keep both and it is closed; keep just one and it is half-open, described as left-open or right-open depending on which end is missing. Intervals can also run forever in one or both directions, written with an infinity symbol, and the whole real line counts as one.
Some edge cases are surprisingly strict. A closed interval must have two finite endpoints, so a ray such as all numbers from zero upward is topologically closed yet is not usually called a closed interval. A single point written as the same number at both ends is a degenerate interval, and the empty set qualifies as an interval too. Any interval that is neither empty nor a single point contains infinitely many numbers.
Because the real numbers have the least-upper-bound property, a non-empty interval is completely fixed by its two endpoints and whether each one is included. Its length is the distance between them, its midpoint the average of the endpoints, and its radius half the length. For an unbounded interval, the size is usually taken to be infinite.
Intervals are everywhere in analysis. The intermediate value theorem says a continuous function maps an interval onto another interval, and integrals of real functions are defined over them. Interval arithmetic computes with ranges instead of single numbers, producing results guaranteed to contain the true answer despite rounding errors and uncertain inputs. The vocabulary is less settled: one standard encyclopedia calls open sets intervals and closed ones segments, while Rudin's classic textbook does the opposite. Countries that use a decimal comma may separate endpoints with a semicolon, and the international standard ISO 31-11 allows a reversed bracket in place of a parenthesis.
Source: Interval (mathematics)
