Basic Analysis I — Real Analysis · Problem 5.2.11

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-integrability

Problem

(Hard) Show that the Thomae or popcorn function (see Example 3.2.12) is Riemann integrable. Therefore, there exists a function discontinuous at all rational numbers (a dense set) that is Riemann integrable. That is, define f:[0,1]→Rf : [0,1] \to \mathbb{R} by f(x):={1/kif x=m/k where m,k∈N and m and k have no common divisors,0if x is irrational.f(x) := \begin{cases} 1/k & \text{if } x = m/k \text{ where } m, k \in \mathbb{N} \text{ and } m \text{ and } k \text{ have no common divisors,} \\ 0 & \text{if } x \text{ is irrational.} \end{cases} Show ∫01f=0\int_0^1 f = 0.

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