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Let f:[0,1]->$\mathbb{R}$ be defined by f(x)=0 when x is irrational and f(x)=1/q when x=p/q with gcd(p,q)=1. Prove that f is integrable on [0,1]. What is $\int_{0}^{1}$f(x) dx? I only have knowledge of Riemann integrability conditions and results.The integral is obviously 0 i think but i really don't know how to do this. Any help would be appreciated!

Fundamentals
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  • See: https://math.stackexchange.com/questions/353452/necessary-and-sufficient-conditions-for-riemann-integrability The given function is continuous almost everywhere (rationals have measure 0) and also is constant outside the rationals. Thus the integral exists and is 0. –  Apr 20 '17 at 10:15

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hint

You prove using the $\delta-\epsilon $ definition that $f $ is continuous at each irrational of $[0,1] $ and discontinuous at every rational.

hamam_Abdallah
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