Basic Analysis I — Real Analysis · Problem 5.1.11

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-sums

Problem

(Challenging) For a bounded function f:[0,1]→Rf : [0,1] \to \mathbb{R}, let Rn:=(1/n)∑i=1nf(i/n)R_n := (1/n)\sum_{i=1}^n f(i/n) (the uniform right-hand rule). a) If ff is Riemann integrable show ∫01f=lim⁡n→∞Rn\int_0^1 f = \lim_{n\to\infty} R_n. b) Find an ff that is not Riemann integrable, but lim⁡n→∞Rn\lim_{n\to\infty} R_n exists.

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