Define $f$ on $[0,1]$ by $f(x)=\begin{cases}x^2 ~~~~~~~~\text{if $x$ is rational}\\ x^3~~~~~~~~\text{if $x$ is irrational}\end{cases}$. Then
- $f$ is not Riemann integrable on $[0,1]$.
- $f$ is Riemann integrable and $\int_{0}^{1}f(x)dx=\frac{1}{4}$,
- $f$ is Riemann integrable and $\int_{0}^{1}f(x)dx=\frac{1}{3}$,
- $\frac{1}{4}=\underline{\int_{0}^{1}}f(x)dx<\overline{\int_{0}^{1}} f(x)dx=\frac{1}{3}$, where $\underline{\int_{0}^{1}}f(x)dx$ and $\overline{\int_{0}^{1}} f(x)dx$ are lower and upper Riemann integral of $f$.
I am facing difficulty in solving the above problem, basically because I haven't solve this kind of question. I tried to see whether it is continuous or monotone, which implies R-integrable. But didn't make much headway in that direction. How to solve this kind of problems? Please give some hints or solutions. Thank you.