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Reference.org
Interval (mathematics)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a real interval is the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of all <a href="/facts/Real_number/R02gw5Pb">real numbers</a> lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative <a href="/facts/Infinity/cF506uD7">infinity</a>, indicating the interval extends without a <a href="/facts/Bounded_set/yCldjtoy">bound</a>. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. </p><p>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 <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a>; the set of all <a href="/facts/Positive_real_numbers/VFXU56H9">positive real numbers</a> is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [<i>a</i>, <i>a</i>]. </p><p>Intervals are ubiquitous in <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>. For example, they occur implicitly in the <a href="/facts/Epsilon-delta_definition_of_continuity/sKbl02pB">epsilon-delta definition of continuity</a>; the <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a> asserts that the image of an interval by a <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> is an interval; <a href="/facts/Integral/V7lyV997">integrals</a> of <a href="/facts/Real_function/bbNG3DnN">real functions</a> are defined over an interval; etc. </p><p><a href="/facts/Interval_arithmetic/1ylnKraf">Interval arithmetic</a> 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 <a href="/facts/Input_data/WQTE8Jas">input data</a> and <a href="/facts/Rounding_error/aF9N8X4I">rounding errors</a>. </p><p>Intervals are likewise defined on an arbitrary <a href="/facts/Total_order/xnTtmuYJ">totally ordered</a> set, such as <a href="/facts/Integers/PUwwrawx">integers</a> or <a href="/facts/Rational_numbers/VJQmyY05">rational numbers</a>. The notation of integer intervals is considered in the special section below. </p>