Let $h: [0,1] \to \Bbb{R}$ be defined by $h(x) = 0$ for x irrational and $h(x) = x^2$ for x rational. Prove $h$ is not integrable using the definition and Riemann's Criterion.
I struggle finding upper partial sums of functions. I know that the inf of the lower sums equal 0 from working a previous version of this ($h(x) = x$ for x rational). But I just cannot wrap my brain around what to do for $x^2$.
Consider the members of the partition that contain a point in $(1/2,1]$. The upper sum for these members is $\gt (1/2)(1/2)^2$, so any upper sum is $\gt 1/8$.