The Dirichlet function $I_Q: \mathbb{R} \to \mathbb{R}$ is defined by
$\qquad I_Q(x) = \begin{cases} 1, & \text{if } x \text{ is rational,} \\ 0, & \text{if } x \text{ is irrational.} \end{cases}$
Prove that $I_Q \notin \mathcal{R}[a, b]$ for any $-\infty < a < b < \infty$.
Hint
Show that for any partition $P$ of $[a, b]$, $U(P, I_Q) = b - a$, $L(P, I_Q) = 0$.