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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 ]
In cases where the integration is permitted to extend over equidistant sections of the interval [,], the composite Boole's rule might be applied. Given N {\displaystyle N} divisions, where N {\displaystyle N} mod 4 = 0 {\displaystyle 4=0} , the integrated value amounts to: [ 4 ]
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.
This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated ...
The function | ′ | is a continuous function from a closed interval [,] to the set of real numbers, thus it is uniformly continuous according to the Heine–Cantor theorem, so there is a positive real and monotonically non-decreasing function () of positive real numbers such that < implies | | ′ (+ ()) | | ′ | | < where = and [,].
For some applications, the integration interval = [,] needs to be divided into uneven intervals – perhaps due to uneven sampling of data, or missing or corrupted data points. Suppose we divide the interval I {\\displaystyle I} into an even number N {\\displaystyle N} of subintervals of widths h k {\\displaystyle h_{k}} .
This method agrees with the definite integral as calculated in more mechanical ways: =. Because the function is continuous and monotonically increasing over the interval, a right Riemann sum overestimates the integral by the largest amount (while a left Riemann sum would underestimate the integral by the largest amount).
Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.