Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals
Let $g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$ where $\chi_{(0,1]} = \left\{\begin{array}{ll} 1&\mbox{if $x-r_n \in (0,1]$,}\\ 0&\mbox{otherwise.} \end{array}\right.$
Show that $g$ is not continuous except possibly on a set of measure $0$.