subinterval

A subinterval is a portion of the real line that itself forms an interval and lies entirely within another interval. In the real line, if I = [a,b] is an interval with a ≤ b, a subinterval J is any interval that satisfies J ⊆ I. Equivalently, J can be described by endpoints c and d with a ≤ c ≤ d ≤ b, and J is one of [c,d], (c,d), [c,d), or (c,d], depending on which endpoints are included. If c = d, J is a degenerate subinterval consisting of a single point.

Examples include [0,1] with subintervals such as [0,1], [0,0], [0,0.5], [0.2,0.8], (0,1), and [0,1). Subintervals can also be used within open parent intervals, such as (a,b) containing subintervals (c,d) with a < c ≤ d < b.

Subintervals inherit basic interval properties: they are themselves intervals, hence connected. Their length is d − c, with the convention that a degenerate subinterval (c = d) has length 0. The inclusion of endpoints affects openness or closedness: a subinterval of a closed interval may be closed, open, or half-open, depending on which endpoints are included. A closed subinterval of a closed interval is compact, and all subintervals of a finite interval have finite length.

Uses of subintervals appear prominently in real analysis and calculus. They are fundamental in partitioning an interval for Riemann sums and definite integrals, and they provide a simple framework for discussing continuity and convergence on restricted domains.