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.
Definitions and terminology
Definition of an interval An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. Examples are the numbers x {\displaystyle x} from one to two, 1 ≤ x ≤ 2 {\displaystyle 1\leq x\leq 2} , and the numbers y {\displaystyle y} greater than 10, i.e. y > 10 {\displaystyle y>10} . Under this definition, the empty set and the entire set of real numbers are both intervals. The endpoints of an interval are its supremum (least upper bound), and its infimum (greatest lower bound), if they exist as real numbers. If the infimum does not exist and the interval is not empty, one says often that the corresponding endpoint is negative infinity, written − ∞ . {\displaystyle -\infty .} Similarly, if the supremum of a non-empty interval does not exist, one says that the corresponding endpoint is positive infinity, written + ∞ . {\displaystyle +\infty .}
Non-empty intervals are completely determined by their endpoints and whether each endpoint belongs to the interval. This is a consequence of the least-upper-bound property of the real numbers, which implies that if the elements of a non-empty interval are all less than some finite value, then the interval has a supremum. This characterization is used to specify intervals by means of interval notation, where a square or rounded bracket (parenthesis) indicates whether or not an endpoint belongs to the inteval.
Open and closed intervals An open interval does not include any endpoint and can be succinctly indicated with parentheses. For example, ( 0 , 1 ) = { x ∣ 0 < x < 1 } {\displaystyle (0,1)=\{x\mid 0<x<1\}} is the interval of all real numbers greater than 0 {\displaystyle 0} and less than 1 {\displaystyle 1} . (This interval can also be denoted by ] 0 , 1 [ {\displaystyle ]0,1[} , see below). The open interval ( 0 , + ∞ ) {\displaystyle (0,+\infty )} consists of real numbers greater than 0 {\displaystyle 0} , i.e., positive real numbers. The open intervals have thus one of the forms
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