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Interval (mathematics)
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{{Short description|All numbers between two given numbers}} {{About|intervals of real numbers and some generalizations|intervals in order theory|Interval (order theory)|other uses|Interval (disambiguation)}} [[File:Interval0.png|thumb|400px|The addition ''x'' + ''a'' on the number line. All numbers greater than ''x'' and less than ''x'' + ''a'' fall within that open interval.]] In [[mathematics]], a '''real interval''' is the [[set (mathematics)|set]] of all [[real number]]s lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative [[infinity]], indicating the interval extends without a [[Bounded set|bound]]. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of {{math|0}}, {{math|1}}, and all numbers in between is an interval, denoted {{math|[0, 1]}} and called the [[unit interval]]; the set of all [[positive real numbers]] is an interval, denoted {{math|(0, β)}}; the set of all real numbers is an interval, denoted {{math|(ββ, β)}}; and any single real number {{mvar|a}} is an interval, denoted {{math|[''a'', ''a'']}}. 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; [[integral]]s of [[real function]]s are defined over an interval; etc. [[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 error]]s. Intervals are likewise defined on an arbitrary [[total order|totally ordered]] set, such as [[integers]] or [[rational numbers]]. The notation of integer intervals is considered [[#Integer intervals|in the special section below]]. {{hatnote|Unless explicitly otherwise specified, all intervals considered in this article are real intervals, that is, intervals of real numbers. Notable generalizations are summarized in a section below possibly with links to separate articles.}}
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