On Riemann Sums

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.