Let $f$ defined on $[0,1]$ by $f(x) = 1/b$ if $x$ can be written as reduced fraction $x=a/b$ for integers $a,b\geqslant1$ and $f(x)=0$ if not. Show that $f$ is integrable.
My attempt was to show that $\bar{S}_{\sigma} (f) \leqslant \underline{S}_{\sigma} (f) + \epsilon$, $\forall \epsilon >0$ and for subdivision $\sigma$. We know as well by density of irrational numbers, that $\underline{S}_{\sigma} (f)=0$ for all subdivision $\sigma$. So i have to show that $\bar{S}_{\sigma} (f) \leqslant \epsilon$, $\forall \epsilon>0$. I noticed as well that there is a finite number such that $1/b > \epsilon$. But, i don't really know how to continue the proof. Thanks in advance
