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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 ] [ 5 ] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [ a , a ] ). [ 6 ]
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
The notation [,) is used to indicate an interval from a to c that is inclusive of —but exclusive of . That is, [ 5 , 12 ) {\displaystyle [5,12)} would be the set of all real numbers between 5 and 12, including 5 but not 12.
In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1].
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
Since () is a sequence of nested intervals, the interval lengths get arbitrarily small; in particular, there exists an interval with a length smaller than . But from s ∈ I n {\displaystyle s\in I_{n}} one gets s − a n < s − σ {\displaystyle s-a_{n}<s-\sigma } and therefore a n > σ {\displaystyle a_{n}>\sigma } .
This notation means “the integral of f(x) from a to b,” and it represents the exact area under the curve f(x) and above the x-axis, between x = a and x = b. The idea behind the Riemann integral is to break the area into small, simple shapes (like rectangles), add up their areas, and then make the rectangles smaller and smaller to get a ...
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .