Let $$f(x) = \begin{cases}x^2 & x\in \Bbb Q \\ x^3 & x \notin \Bbb Q \end{cases}$$
I want to show that this function is not Riemann integrable on $[0,1].$
I though of find the lower and upper $R$-integral and show them not equal but but with functions like Dirichlet function, it is easy as function take value 0 on rational and $1$ on irrational. Here it is not that clear.
Suppose P is my partition here, P= $\{x_1,x_2, \dots\ x_n\}$, so for $U(f,P)$ is need maximum in intervals like $[x_{i-1},x_i]$ and for $L(f,P)$, I need minimum. But how can I know where will it be maximum or minimum. If points of my partition are rational then maximum will be $x_{i-1}^2$ and minimum will be $x_i^2$. But I dont seem to be getting anywhere like that.