$f:[0,1] \to \mathbb R$ be a function such that $f(x)=x^2 , \forall x \in \mathbb Q ; f(x)=x^3$ ; what is the upper and lower integral of $f$?

Question

Let $f:[0,1] \to \mathbb R$ be a function such that $f(x)=x^2 , \forall x \in \mathbb Q ; f(x)=x^3 , \forall x \notin \mathbb Q$ . Then is it true that the lower integral of $f$ over $[0,1]$ is $1/4$ and the upper integral of $f$ over $[0,1]$ is $1/3$ ?

Answer 1

Since on the unit interval $x^3\leq x^2$, the lower integral coincides with the integral of $x^3$, and the upper integral coincides with the integral of $x^2$, so your answers are correct. This relies on the fact that both rational numbers and irrational numbers are dense subsets.