Show that there is a sequence $(P_n)$ of partitions of $[a,b]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Question

Let $g(x)= \begin{cases} 0, & \text{if }x\in\mathbb{Q} \\ 1/x, & \text{if }x\not\in\mathbb{Q} \end{cases}$, $x\in[0,1]$. Show that $\exists$ sequence $(P_n)$ of tagged partitions of $[a,b]$ such that $||P_n||\to0$ & $\lim\limits_{n\to\infty} S(g,P_n)$ exists.

Why is $g\not\in\mathscr{R}[0,1]$?

I don't really know where to start.

Answer 1

$(1)$ Choose as tags rational numbers, and any sufficiently small partition.

$(2)$ Is $g$ bounded on $[0,1]$?