Interval

An interval is a set of real numbers that contains every number between any two of its points. On the real number line, intervals are connected and convex, meaning that for any x and y in the interval and any t with 0 ≤ t ≤ 1, the point (1−t)x + ty lies in the interval.

Common forms are distinguished by endpoint inclusion. An open interval (a, b) contains all x with a < x < b. A closed interval [a, b] contains all x with a ≤ x ≤ b. Half-open or half-closed intervals include one endpoint: [a, b) or (a, b]. Intervals can also be unbounded: (a, ∞), [a, ∞), (−∞, b], and (−∞, ∞). A degenerate interval [a, a] consists of a single point. The empty interval is sometimes defined as ∅, typically arising from a > b, though some authors reserve that word for the empty set rather than an interval.

Interval notation succinctly records endpoint inclusion with brackets ( [ or ] ) and parentheses ( ( or ) ). For finite intervals, the length is b − a when a < b; for unbounded intervals, the length is infinite.

Key properties include that the intersection of two intervals is either an interval or empty, while the union of two intervals may or may not be an interval depending on whether they overlap or touch. In real analysis, intervals form a fundamental class of sets on the real line and play a central role in defining domains, continuity, and convergence. Intervals also appear in other contexts, such as time spans and musical or perceptual scales, where they denote ranges of values.