In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. An interval can contain neither endpoint, either endpoint, or both endpoints.
For example, the set of real numbers consisting of,, 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 ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, 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.
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 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 interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset.
The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers.[1] If the infimum does not exist, one says often that the corresponding endpoint is
-infty.
+infty.
Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the least-upper-bound property of the real numbers. This characterization is used to specify intervals by mean of , which is described below.
An does not include any endpoint, and is indicated with parentheses.[2] For example,
(0,1)=\{x\mid0<x<1\}
\begin{align} (a,b)&=\{x\inR\mida<x<b\},\\ (-infty,b)&=\{x\inR\midx<b\},\\ (a,+infty)&=\{x\inR\mida<x\},\\ (-infty,+infty)&=\R, \end{align}
a
b
a\leb.
a=b
(a,a)=\varnothing,
A is an interval that includes all its endpoints and is denoted with square brackets. For example, means greater than or equal to and less than or equal to . Closed intervals have one of the following forms in which and are real numbers such that
a\leb\colon
[a,b]=\{x\inR\mida\lex\leb\}
\R
A has two endpoints and includes only one of them. It is said 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.[3] 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
\begin{align} \left(a,b\right]&=\{x\in\R\mida<x\leb\},\\ \left[a,b\right)&=\{x\in\R\mida\lex<b\},\\ \left[a,+infty\right)&=\{x\in\R\mida\lex\},\\ \left(-infty,b\right]&=\{x\in\R\midx\leb\}. \end{align}
(-infty,b]
[a,+infty)
(a,b]
[a,b)
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.[4]
A is any set consisting of a single real number (i.e., an interval of the form).[5] 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.
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.
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 empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are open sets of the real line in its standard topology, and form a base of the open sets.
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. So, the closed intervals coincide with the closed sets in that topology.
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 .
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 .
However, 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[6] 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[7] 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.
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.
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,
\begin{align} (a,b)=en{]}a,bose{[}&=\{x\in\R\mida<x<b\},\\[5mu] [a,b)=en{[}a,bose{[}&=\{x\in\R\mida\lex<b\},\\[5mu] (a,b]=en{]}a,bose{]}&=\{x\in\R\mida<x\leb\},\\[5mu] [a,b]=en{[}a,bose{]}&=\{x\in\R\mida\lex\leb\}. \end{align}
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.[8] 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 .
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
R+.
\R
When and are integers, the notation ⟦a, b⟧, or or or just, is 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