Let $\displaystyle f: [0,1] \rightarrow \mathbb{R}$ given by
$f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \\ 0 & x = 0 \\ \\ \frac{1}{q_x} & x = \frac{p_x}{q_x} \in \mathbb{Q} \backslash \{0\}, \ p_x \in \mathbb{Z}, \ q_x \in \mathbb{N}, \ \text{gcd}(|p_x|, q_x) = 1 \end{cases}$
Is $\displaystyle f$ Riemann integrable?
I am trying to use the equivalent statements $\displaystyle g:[a,b] \rightarrow \mathbb{R}$ is Riemann integrable and $\displaystyle \forall \epsilon >0 \ \exists$ step functions $\displaystyle \rho, \psi$ with $\displaystyle \rho \leq g \leq \psi$ such that $\displaystyle \int_a^b (\psi - \rho) \leq \epsilon$.
I guess that means I would have to somehow show that given $\displaystyle \epsilon$, there exists only a finite amount of $\displaystyle x \in [0,1]$ with $\displaystyle f(x) \geq \epsilon$?
Is it recommended that I consider something else instead? If not, how should I do this?