Q: Positive continuous $f_n:\mathbb{R}\to\mathbb{R}$ such that $\{f_n(x)\}$ is unbounded $\iff$ $x\in\mathbb{Q}$. 
Does there exist a sequence of continuous positive functions $f_n$ on $\mathbb{R}$ such that the sequence $\{f_n(x)\}_{n=1}^\infty$ is unbounded if and only if $x\in\mathbb{Q}$?

I was thinking of letting $\mathbb{Q}=\{r_n\}_{n=1}^\infty$ and for each $n$, letting $f_n\in C(\mathbb{R})$ be such that $f_n(x)=n$ for $x=r_1,\ldots,r_n$ and $0$ outside small disjoint intervals around $r_1,\ldots,r_n$. We would have $f_n(x)\to\infty$ for all $x\in\mathbb{Q}$. But maybe $f_n(x)$ is not necessarily bounded for $x\notin\mathbb{Q}$.
