See the paper, First Class Functions, in the American Mathematical Monthly 98 (March, 1991) 237-240.
EDIT: Let fn(x)=nfn(x)=n if x=p/qx=p/q with qlenqlen, fn(x)=0fn(x)=0 if x=(p/q)pmn−4x=(p/q)pmn−4 with qlenqlen, and let fnfn be piecewise linear between these points. So fnfn is continuous, mostly zero, but with a sharp spike at each rational. Clearly fn(x)fn(x) goes to infinity with nn at all rational xx. If xx is irrational and has only finitely many rational approximations p/qp/q such that |x−(p/q)|leq−4|x−(p/q)|leq−4 (and this is all xx save a set of measure zero), then fn(x)=0fn(x)=0 for all nn sufficiently large. If xx has infinitely many rational approximations with |x−(p/q)|leq−4|x−(p/q)|leq−4, then fn(x)=0fn(x)=0 for most nn (those that are far from a qq which gives a good approximation, and those qq are guaranteed to be few and far between), but is occasionally quite large, so fn(x)fn(x) has no limit, finite or infinite.
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