See the paper, First Class Functions, in the American Mathematical Monthly 98 (March, 1991) 237-240.

EDIT: Let $f_n(x)=n$ if $x=p/q$ with $q\le n$, $f_n(x)=0$ if $x=(p/q)\pm n^{-4}$ with $q\le n$, and let $f_n$ be piecewise linear between these points. So $f_n$ is continuous, mostly zero, but with a sharp spike at each rational. Clearly $f_n(x)$ goes to infinity with $n$ at all rational $x$. If $x$ is irrational and has only finitely many rational approximations $p/q$ such that $|x-(p/q)|\le q^{-4}$ (and this is all $x$ save a set of measure zero), then $f_n(x)=0$ for all $n$ sufficiently large. If $x$ has infinitely many rational approximations with $|x-(p/q)|\le q^{-4}$, then $f_n(x)=0$ for most $n$ (those that are far from a $q$ which gives a good approximation, and those $q$ are guaranteed to be few and far between), but is occasionally quite large, so $f_n(x)$ has no limit, finite or infinite.