real analysis - $L^1$ function unbounded on every interval

[My question concerns part of Exercise 2.25 in Folland's Real Analysis text.]

I'm looking at the function $g(x)=\sum_{n=1}^\infty 2^{-n}f(x-r_n)$, where $f(x)=x^{-1/2}$ for $x\in (0,1)$ and $f(x)=0$ elsewhere, and $\{r_n\}$ is some enumeration of the rational numbers. I am trying to prove that $g$ is discontinuous at every point. It is easy enough to see that $g$ is discontinuous wherever it is finite (since it is unbounded on every interval), and that it is finite almost everywhere (since it is in particular in $L^1$). However, it seems to me that to show that $g$ is discontinuous at a point $x_0$ with $g(x_0)=+\infty$, one would have to exhibit a sequence $\{x_n\}$ converging to $x_0$ such that $\{g(x_n)\}$ is bounded. It seems intuitively obvious to me that such a sequence exists, but I have not been able to construct one. Any suggestions? (Hints preferred to answers.)