$f(x)= 0$ , if $x \notin \mathbb{Q}$, otherwise $f(x)=1/q$ for $x=p/q$ such that $p$ and $q$ don't share common divisor. I 'd love your help proving that $f$ is integrable and that $\int_{0}^{1}f=0$.
I showed that the lower Darboux sum is $0$ and I basically need to show that for every epsilon we can find division such that the upper Darboux sum is smaller than the given epsilon.
The upper darboux sum is $\bar{S}=\sum_{1}^{n}f(x_i) \Delta x_i$ for all $x_i$ of the partition, I tried to replace the $f(x_i)$ in $1/q_i$, and check if this series converges to $0$.
Thanks a lot.