Interval (mathematics)

This article is about intervals of real numbers and some generalizations. For intervals in order theory, see Interval (order theory). For other uses, see Interval (disambiguation).

In mathematics, an interval is the set of all real numbers lying between two fixed endpoints with no "gaps". For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval. An interval may contain neither endpoint (called an open interval), both endpoints (called a closed interval), or either endpoint (called a semi-open or semi-closed interval).

The intervals just described are the bounded intervals. Often intervals are also allowed to extend without bound in one or both directions, with the unbounded side being denoted by a positive or negative infinity symbol. The set of all positive real numbers is an interval in this sense, denoted (0, ∞); the set of all real numbers is an interval that is unbounded on both ends, denoted (−∞, ∞).

Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. For example, interval arithmetic consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of input data and rounding errors.

Intervals can be defined more generally on any totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.

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