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**Opalg** Let $\displaystyle U_n = \{x\in X : \exists\text{ op{}en }V_x\ni x \text{ such that }v,w\in V_x\,\Rightarrow\,|f(v)-f(w)|<1/n\}.$

If $\displaystyle x\in U_n$ then $\displaystyle V_x\subseteq U_n$, so $\displaystyle U_n$ is open. Also, f is continuous at x if and only if $\displaystyle x\in\bigcap_nU_n$. Thus the set of points of continuity of f is a $\displaystyle G_\delta$.

What bothers me about that argument is that it seems to apply to an arbitrary function f. (It doesn't assume that f is the pointwise limit of a sequence of continuous functions.) Am I missing something?