Riemann integrability

[Tolaso J Kos]

Let $f:[0, 1] \rightarrow \mathbb{R}$ be defined as:

$$f(x)= \left\{\begin{matrix}
0&, &x \in [0,1]\cap \left ( \mathbb{R} \setminus \mathbb{Q} \right ) \\
x_n &, &x=q_n \in [0,1] \cap \mathbb{Q} \\
\end{matrix}\right.$$


where $x_n$ is a sequence such that $\lim x_n =0$ and $0\leq x_n \leq 1$ and $q_n$ is an enumeration of the rationals of the interval $[0, 1]$.

Prove that $f$ is Riemann integrable and that $\displaystyle \int_0^1 f(x)\, {\rm d}x =0$.