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On Riemann Sums
Brian S. Thomson
Source: Real Anal. Exchange Volume 37, Number 1
(2011), 221-242.
Abstract
If a sum of the form \[\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\] is used without the familiar requirement that the sequence of points \(a=x_0, x_1, \dots, x_n=b\) is increasing, do we still get a useful approximation to the integral? With a suitable set of hypotheses the answer is yes. We give applications to change of variable formulas and the problem of characterizing derivatives.
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Keywords: Riemann sums; Riemann integral; Lebesgue integral; Stieltjes integral; Henstock-Kurzweil integral; change of variable; derivative
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.rae/1335806774
Zentralblatt MATH identifier: 06038700
Mathematical Reviews number (MathSciNet): MR3016862
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Zentralblatt MATH: 1018.01004
Digital Object Identifier: doi:10.1016/S0723-0869(01)80019-0
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