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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.

First Page:
Primary Subjects: 26A24, 26A42
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1335806774
Zentralblatt MATH identifier: 06038700
Mathematical Reviews number (MathSciNet): MR3016862

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