Show $f(x)=x\cdot 1_{\mathbb R\setminus \mathbb Q}(x)$ is a limit of Riemann-Integrable functions?

Question

Suppose

$f(x)=\begin{cases}x&\mbox{if }x\notin Q\\0&\mbox{if }x\in Q\end{cases}$

(where Q means rational numbers, and not in Q means irrational numbers).

How do you show $f(x)$ is a limit of a sequence of Riemann-Integrable functions?

Answer 1

Arrange rational numbers in a sequence $\{r_1,r_2,\cdots \}$. Let $f_n(x)=x$ if $x \notin \{r_1,r_2,\cdots,r_n\}$ and $0$ otherwise. Since $f_n$ has only finite number of discontinuities it is Riemann integrable. Clearly, $f_n (x) \to f(x)$ for all $x$.