The function is given by $$f(x)=sin(1/x)$$ if $x$ is irrational and $$f(x)=0$$ when $x$ is rational and $x$ belongs to $[0,1]$.
I have a feeling that such a highly discontinuous functions can't be Riemann integrated. I was trying to prove it using upper and lower sums. In any partition of the interval $[0,1]$ each sub interval will have rational and irrational numbers. So $$L(f)=sup(L(f,P))$$ is 0. But, what about $U(f)$. How do I find the maximum of the function at each of the sub intervals of the partition?