See the paper, First Class Functions, in the American Mathematical Monthly 98 (March, 1991) 237-240.
EDIT: Let fn(x)=n if x=p/q with qlen, fn(x)=0 if x=(p/q)pmn−4 with qlen, and let fn be piecewise linear between these points. So fn is continuous, mostly zero, but with a sharp spike at each rational. Clearly fn(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)|leq−4 (and this is all x save a set of measure zero), then fn(x)=0 for all n sufficiently large. If x has infinitely many rational approximations with |x−(p/q)|leq−4, then fn(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 fn(x) has no limit, finite or infinite.
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