Open Access
May 2014 Rates of Uniform Convergence for Riemann Integrals
J. Alan Alewine
Missouri J. Math. Sci. 26(1): 48-56 (May 2014). DOI: 10.35834/mjms/1404997108


A function $f \colon [0,1]\to {\mathbb R}$ is Riemann integrable if and only if its Riemann sums $f(T)$ and $f(T')$ get closer to each other as $\delta\to 0$, uniformly over all $\delta$-fine tagged divisions $T$ and $T'$. We show that $\delta^{-1}|f(T)-f(T')|\asymp \mbox{Var}(f)$. We also give an example of a function $f\notin \mbox{BV}$ with $|f(T)-f(T')|= {\mathcal O}(\delta|\ln \delta|)$. As a lemma, we show that any $f\in\mbox{BV}$ can be approximated uniformly by a step function $g$ with $\mbox{Var}(g)\approx\mbox{Var}(f)$.


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J. Alan Alewine. "Rates of Uniform Convergence for Riemann Integrals." Missouri J. Math. Sci. 26 (1) 48 - 56, May 2014.


Published: May 2014
First available in Project Euclid: 10 July 2014

zbMATH: 1298.26028
MathSciNet: MR3263541
Digital Object Identifier: 10.35834/mjms/1404997108

Primary: 26A42
Secondary: 26A45

Keywords: Convergence rates , Riemann integral , Uniform convergence

Rights: Copyright © 2014 Central Missouri State University, Department of Mathematics and Computer Science

Vol.26 • No. 1 • May 2014
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