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47-501: [REDACTED] Look up endpoint in Wiktionary, the free dictionary. An endpoint , end-point or end point may refer to: Endpoint (band) , a hardcore punk band from Louisville, Kentucky Endpoint (chemistry) , the conclusion of a chemical reaction, particularly for titration Outcome measure , a measure used as an endpoint in research Clinical endpoint , in clinical research,

94-438: A , b ) {\displaystyle (a,b)} is a 1-dimensional open ball with a center at 1 2 ( a + b ) {\displaystyle {\tfrac {1}{2}}(a+b)} and a radius of 1 2 ( b − a ) . {\displaystyle {\tfrac {1}{2}}(b-a).} The closed finite interval [ a , b ] {\displaystyle [a,b]}

141-406: A , b ) {\displaystyle [a,b)} are neither an open set nor a closed set. If one allows an endpoint in the closed side to be an infinity (such as (0,+∞] ), the result will not be an interval, since it is not even a subset of the real numbers. Instead, the result can be seen as an interval in the extended real line , which occurs in measure theory , for example. In summary,

188-450: A and b are real numbers such that a ≤ b : {\displaystyle a\leq b\colon } The closed intervals are those intervals that are closed sets for the usual topology on the real numbers. The empty set and R {\displaystyle \mathbb {R} } are the only intervals that are both open and closed. A half-open interval has two endpoints and includes only one of them. It

235-448: A and b included. The notation [ a .. b ] 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

282-493: A half-space is taken as a kind of degenerate ball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint. A finite interval is (the interior of) a 1-dimensional hyperrectangle . Generalized to real coordinate space R n , {\displaystyle \mathbb {R} ^{n},} an axis-aligned hyperrectangle (or box)

329-419: A ,  a ) , [ a ,  a ) , and ( a ,  a ] represents the empty set , whereas [ a ,  a ] denotes the singleton set  { a } . When a > b , 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 ( a , b ) is often used to denote an ordered pair in set theory,

376-491: A , +∞) 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, (0, +∞) is the set of positive real numbers , also written as R + . {\displaystyle \mathbb {R} _{+}.} The context affects some of

423-456: A disease, symptom, or sign that constitutes one of the target outcomes of the trial or its participants In mathematics [ edit ] Endpoint, the lower or upper bound of an interval (mathematics) Endpoint, either of the two nodes of an edge in a graph Endpoint, either of two extreme points on a line segment Endpoint, either of two extreme points on a curve In computing [ edit ] Communication endpoint ,

470-456: A disease, symptom, or sign that constitutes one of the target outcomes of the trial or its participants In mathematics [ edit ] Endpoint, the lower or upper bound of an interval (mathematics) Endpoint, either of the two nodes of an edge in a graph Endpoint, either of two extreme points on a line segment Endpoint, either of two extreme points on a curve In computing [ edit ] Communication endpoint ,

517-454: A finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a .. b  − 1  , a  + 1 .. b  , or a  + 1 .. b  − 1 . Alternate-bracket notations like [ a .. b ) or [ a .. b [ are rarely used for integer intervals. The intervals are precisely the connected subsets of R . {\displaystyle \mathbb {R} .} It follows that

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564-410: 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 I is the largest open interval that is contained in I ; it is also the set of points in I which are not endpoints of I . The closure of I is the smallest closed interval that contains I ; which

611-441: 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. A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [ a , a ] ). 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

658-415: A subset X ⊆ R {\displaystyle X\subseteq \mathbb {R} } is also the convex hull of X . {\displaystyle X.} 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

705-456: Is a closed set of the real line , but an interval that is a closed set need not be a closed interval. For example, intervals ( − ∞ , b ] {\displaystyle (-\infty ,b]} and [ a , + ∞ ) {\displaystyle [a,+\infty )} are also closed sets in the real line. Intervals ( a , b ] {\displaystyle (a,b]} and [

752-448: Is a bounded real interval whose endpoints are j 2 n {\displaystyle {\tfrac {j}{2^{n}}}} and j + 1 2 n , {\displaystyle {\tfrac {j+1}{2^{n}}},} where j {\displaystyle j} and n {\displaystyle n} are integers. Depending on the context, either endpoint may or may not be included in

799-516: Is a connected subset.) In other words, we have The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example ( a , b ) ∪ [ b , c ] = ( a , c ] . {\displaystyle (a,b)\cup [b,c]=(a,c].} If R {\displaystyle \mathbb {R} }

846-411: Is a degenerate interval (see below). The open intervals are those intervals that are open sets for the usual topology on the real numbers. A closed interval is an interval that includes all its endpoints and is denoted with square brackets. For example, [0, 1] means greater than or equal to 0 and less than or equal to 1 . Closed intervals have one of the following forms in which

893-441: Is also the set I augmented with its finite endpoints. For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X , and does not properly contain any other interval that also contains X . An interval I is a subinterval of interval J if I is a subset of J . An interval I is a proper subinterval of J if I is a proper subset of J . However, there

940-449: 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

987-452: 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. If the infimum does not exist, one says often that the corresponding endpoint is − ∞ . {\displaystyle -\infty .} Similarly, if

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1034-551: Is described below. An open interval does not include any endpoint, and is 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 and less than 1 . (This interval can also be denoted by ]0, 1[ , see below). The open interval (0, +∞) consists of real numbers greater than 0 , i.e., positive real numbers. The open intervals are thus one of

1081-434: 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

1128-405: 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. For example, (0, 1] means greater than 0 and less than or equal to 1 , while [0, 1) means greater than or equal to 0 and less than 1 . The half-open intervals have the form Every closed interval

1175-414: 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

1222-740: Is the Cartesian product of n {\displaystyle n} finite intervals. For n = 2 {\displaystyle n=2} this is a rectangle ; for n = 3 {\displaystyle n=3} this is a rectangular cuboid (also called a " box "). Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any n {\displaystyle n} intervals, I = I 1 × I 2 × ⋯ × I n {\displaystyle I=I_{1}\times I_{2}\times \cdots \times I_{n}}

1269-404: Is the corresponding closed ball, and the interval's two endpoints { a , b } {\displaystyle \{a,b\}} form a 0-dimensional sphere . Generalized to n {\displaystyle n} -dimensional Euclidean space , a ball is the set of points whose distance from the center is less than the radius. In the 2-dimensional case, a ball is called a disk . If

1316-407: 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

1363-484: Is viewed as a metric space , its open balls are the open bounded intervals  ( c  +  r ,  c  −  r ) , and its closed balls are the closed bounded intervals  [ c  +  r ,  c  −  r ] . In particular, the metric and order topologies in the real line coincide, which is the standard topology of the real line. Any element  x of an interval  I defines a partition of  I into three disjoint intervals I 1 ,  I 2 ,  I 3 : respectively,

1410-408: 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 ] a , b [ to denote the open interval. The notation [ a , b ] too is occasionally used for ordered pairs, especially in computer science . Some authors such as Yves Tillé use ] a , b [ to denote the complement of

1457-494: 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 , Each interval (

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1504-616: 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

1551-428: The above definitions and terminology. For instance, the interval (−∞, +∞)  =  R {\displaystyle \mathbb {R} } is closed in the realm of ordinary reals, but not in the realm of the extended reals. When a and b are integers , the notation ⟦ a, b ⟧, or [ a .. b ] or { a .. b } or just a .. b , is sometimes used to indicate the interval of all integers between

1598-438: The elements of  I that are less than  x , the singleton  [ x , x ] = { x } , {\displaystyle [x,x]=\{x\},} and the elements that are greater than  x . The parts I 1 and I 3 are both non-empty (and have non-empty interiors), if and only if x is in the interior of  I . This is an interval version of the trichotomy principle . A dyadic interval

1645-499: The entity on one end of a transport layer connection Endpoint (web API) , a function or procedure call that is part of an API in software engineering Endpoint security , the security model around end user devices such as PCs, laptops and mobile phones See also [ edit ] Enden Point , in Antarctica Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1692-439: The entity on one end of a transport layer connection Endpoint (web API) , a function or procedure call that is part of an API in software engineering Endpoint security , the security model around end user devices such as PCs, laptops and mobile phones See also [ edit ] Enden Point , in Antarctica Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1739-412: 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 a and b , including a and b , is often denoted [ a ,  b ] . The two numbers are called

1786-453: The forms where a {\displaystyle a} and b {\displaystyle b} are real numbers such that a ≤ b . {\displaystyle a\leq b.} When a = b {\displaystyle a=b} in the first case, the resulting interval is the empty set ( a , a ) = ∅ , {\displaystyle (a,a)=\varnothing ,} which

1833-432: The 💕 [REDACTED] Look up endpoint in Wiktionary, the free dictionary. An endpoint , end-point or end point may refer to: Endpoint (band) , a hardcore punk band from Louisville, Kentucky Endpoint (chemistry) , the conclusion of a chemical reaction, particularly for titration Outcome measure , a measure used as an endpoint in research Clinical endpoint , in clinical research,

1880-407: The image of an interval by any continuous function from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } is also an interval. This is one formulation of the intermediate value theorem . The intervals are also the convex subsets of R . {\displaystyle \mathbb {R} .} The interval enclosure of

1927-405: The interval  ( a ,  b ) ; namely, the set of all real numbers that are either less than or equal to a , or greater than or equal to b . 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 [−∞,  b ]  , (−∞,  b ]  , [ a , +∞]  , and [

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1974-439: The interval. Dyadic intervals have the following properties: The dyadic intervals consequently have a structure that reflects that of an infinite binary tree . Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement , multigrid methods and wavelet analysis . Another way to represent such a structure is p-adic analysis (for p = 2 ). An open finite interval (

2021-437: 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 ; 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 [ a , a ] . Intervals are ubiquitous in mathematical analysis . For example, they occur implicitly in

2068-575: The size of the empty interval may be defined as 0 (or left undefined). The centre ( midpoint ) of a bounded interval with endpoints a and b is ( a  +  b )/2 , and its radius is the half-length | a  −  b |/2 . 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 [0, 1) = { x | 0 ≤ x < 1} , for example,

2115-423: The supremum does not exist, one says that the corresponding endpoint is + ∞ . {\displaystyle +\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 interval notation , which

2162-458: The title Endpoint . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Endpoint&oldid=1224419662 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages endpoint From Misplaced Pages,

2209-468: The title Endpoint . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Endpoint&oldid=1224419662 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Interval (mathematics) For example,

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