Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff. Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below. An does not include its endpoints, and is indicated with parentheses. For example, means greater than 0 and less than 1. This means = . This interval can also be denoted by ]0, 1[, see below. A is an interval which includes all its limit points, and is denoted with square brackets. For example, means greater than or equal to 0 and less than or equal to 1. A includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. For example, means greater than 0 and less than or equal to 1, while means greater than or equal to 0 and less than 1. A is any set consisting of a single real number (i.e., an interval of the form ). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements.