Is $f$ riemann integrable on $[0, 1]$?

Question

If $f(x) = \begin{cases} 0 & \text{ if }x\in\Bbb R\setminus\Bbb Q\\ 1/q & \text{ if } x = p/q;~p,q\in\Bbb Z,~q\ne 0,~\gcd(p, q)=1 \end{cases}$

Is $f$ Riemann integrable on $[0, 1]$?

Welcome to math stackexchange! Thi is a common problem, so we would greatly appreciate it if you told us a little about what you know how to do,and what you have tried on this problem. — Stella Biderman, Dec 09 '15 at 21:32
Relevant keywords: Thomae function, or "popcorn function." (This should generate quite a few matches in the search word, rightmost corner of your screen) — Clement C., Dec 09 '15 at 21:32
you can use Lebesgue criterion for Riemann Integrability, just find points of discontinuity and show that its measure is $0$ — Kerr, Dec 09 '15 at 21:34
So many downvotes for a newcomer... That's not a warm and fair welcome! — mathcounterexamples.net, Dec 09 '15 at 21:51

score 0 · Answer 1 · answered Dec 09 '15 at 21:31

0

Hint: In which points is your $f$ continuous?

answered Dec 09 '15 at 21:31

gerw

It's continuous on the irrationals and integers, if I'm correct. – Dminus Dec 11 '15 at 15:12
It is not continuous at the integers. Consider $1/n \to 0$. – gerw Dec 11 '15 at 16:22
So then if it's not continuous on the integers it can't be riemann integrable? – Dminus Dec 11 '15 at 17:27
Continuity at integers is not at all related to Riemann integrability. – gerw Dec 11 '15 at 17:50

1 Answers1