![]() If the infimum does not exist, one says often that the corresponding endpoint is − ∞. The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers. Definitions and terminology Īn interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. Notable generalizations are summarized in a section below possibly with links to separate articles. Unless explicitly otherwise specified, all intervals considered in this article are real intervals, that is, intervals of real numbers. The notation of integer intervals is considered in the special section below. Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. 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. 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. Intervals are ubiquitous in mathematical analysis. An interval can contain neither endpoint, either endpoint, or both endpoints.įor example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted and called the unit interval the set of all positive real numbers 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. Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. In mathematics, a ( real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps". All numbers greater than x and less than x + a fall within that open interval. For other uses, see Interval (disambiguation). The range is an important concept in mathematics as it helps in determining the spread of the data and provides an insight into the variability of the data. ![]() In mathematics, range refers to the set of all output values of a function.It is the difference between the largest and smallest numbers in a set or sequence. For intervals in order theory, see Interval (order theory). Understanding the definition of range in math. This article is about intervals of real numbers and some generalizations.
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