Rates of Uniform Convergence for Riemann Integrals

Abstract

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 $\mbox{Var}(g)\approx\mbox{Var}(f)$.