In mathematics, an interval is the set of all real numbers lying between two fixed lower or upper bounds ("endpoints") with no "gaps". For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted 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 ; 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 from one to two, , and the numbers greater than 10, i.e. . 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 Similarly, if the supremum of a non-empty interval does not exist, one says that the corresponding endpoint is positive infinity, written
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 , where a square or rounded bracket (parenthesis) indicates whether or not an endpoint belongs to the inteval.
Open and closed intervals An does not include any endpoint and can be succinctly indicated with parentheses. For example, is the interval of all real numbers greater than and less than . (This interval can also be denoted by , see below). The open interval consists of real numbers greater than , i.e., positive real numbers. The open intervals have thus one of the forms
where and are real numbers such that In the last case, the resulting interval is the empty set and does not depend on . The open intervals are those intervals that are open sets for the usual topology on the real numbers, and they form a base of the open sets.
A is an interval which includes both endpoints, which are finite. A closed interval is denoted with square brackets. For example, is the closed interval with contents greater than or equal to and less than or equal to . Closed intervals are by definition non-empty. Thus every closed interval has the form
where . The interval consisting of a single point is sometimes called a degenerate closed interval.
In addition to the closed intervals common in analysis, unbounded intervals that include their finite endpoint such as or are topologically closed (meaning that they contain all of their boundary points that are real numbers) but are not usually called "closed intervals" in analysis, that term being reserved for the closed and bounded case. The (bounded) closed intervals together with the semi-infinite closed intervals, and the interval comprise those intervals that are closed sets for the usual topology on the real numbers.
Half-open intervals A has two distinct finite endpoints, and includes one but not the other. It is said to be left-open or right-open depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals. For example, means greater than and less than or equal to , while means greater than or equal to and less than . The half-open intervals have the form
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. The only intervals that appear twice in the above classification are and that are both open and closed. See Definition 9.1.1.
Degenerate intervals 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 , and has infinitely many elements.
Bounded intervals An interval is said to be left-bounded or right-bounded, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite intervals.
Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as , and the size of the empty interval may be defined as (or left undefined).
The centre (midpoint) of a bounded interval with endpoints and is , and its radius is the half-length . These concepts are undefined for empty or unbounded intervals.
Categorisation by minimum and maximum elements An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it contains neither. The interval , for example, is left-closed and right-open. The set of non-negative reals is a closed interval that is right-open but not left-open.
An interval is said to be left-closed if it has a minimum element or is left-unbounded, right-closed if it has a maximum or is right unbounded; it is simply closed if it is both left-closed and right closed.
Sub-intervals and related constructions An interval is a subinterval of interval if is a subset of . An interval is a proper subinterval of if is a proper subset of .
The interior of an interval is the largest open interval that is contained in ; it is also the set of points in which are not endpoints of . The closure of is the smallest closed interval that contains ; which is also the set augmented with its finite endpoints.
For any set of real numbers, the interval enclosure or interval span of is the unique interval that contains , and does not properly contain any other interval that also contains .
Segments and intervals There is conflicting terminology for the terms segment and interval, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The Encyclopedia of Mathematics defines interval (without a qualifier) to exclude both endpoints (i.e., open interval) and segment to include both endpoints (i.e., closed interval), while Rudin's Principles of Mathematical Analysis calls sets of the form [a, b] intervals and sets of the form (a, b) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open, closed, or half-open), regardless of whether endpoints are included.
Notations for intervals The interval of numbers between and , including and , is often denoted . The two numbers are called the endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be used as a separator to avoid ambiguity.
Including or excluding endpoints To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11. Thus, in set builder notation,
Each interval , , and represents the empty set, whereas denotes the singleton set . When , all four notations are usually taken to represent the empty set.
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. That is why Bourbaki introduced the notation to denote the open interval. The notation too is occasionally used for ordered pairs, especially in computer science.
Some authors such as Yves Tillé use to denote the complement of the interval ; namely, the set of all real numbers that are either less than or equal to , or greater than or equal to .
Infinite endpoints In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with and .
In this interpretation, the notations , , , and are all meaningful and distinct. In particular, denotes the set of all ordinary real numbers, while denotes the extended reals.
Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction. For example, is the set of positive real numbers, also written as The context affects some of the above definitions and terminology. For instance, the interval = is closed in the realm of ordinary reals, but not in the realm of the extended reals.
Integer intervals When and are integers, the notations ⟦a, b⟧, , , or just , are sometimes used to indicate the interval of all integers between and included. The notation is used in some programming languages; in Pascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array.
Another way to interpret integer intervals are as sets defined by enumeration, using ellipsis notation.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing , , or . Alternate-bracket notations like or are rarely used for integer intervals.
Properties The intervals are precisely the connected subsets of It follows that the image of an interval by any continuous function from to is also an interval. This is one formulation of the intermediate value theorem.
The intervals are also the convex subsets of The interval enclosure of a subset is also the convex hull of
The closure of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every connected subset of a topological space is a connected subset.) In other words, we have