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Let $f:[0,1]\to R$ be a function s.t. $f=0$ if $x=0$ or $x\in (R-Q)\cap[0,1]$ and $f={1\over q}$ if $x\in Q \cap[0,1]$ and $x={p\over q}$ for positive integer $s,t$ without common factor. Is $f$ integrable??

The definition of integrability is : a function is integrable if

(i) Both $L(f)$ and $U(f)$ are well defined and $L(f)=U(f)$ where $L(f)$ is the supremum of lower darboux sum and $U(f)$ is the infimum of the upper darboux sum

ii)For all $\epsilon$, there exist $\delta>0$ s.t. $\max(P)<\delta\implies U(f,P)-L(f,P)<\epsilon$ where P is a partition of the domain.

johnny
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    Yes, it is. See this post to see why $U(f)=0$. It is easy to show that $L(f)=0$. (You could also appeal to the theorem that a bounded function is Riemann integrable over $[a,b]$ if and only if its set of discontinuities in $[a,b]$ has measure zero.) – David Mitra May 24 '13 at 15:06

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