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Suppose we have a set $A\subset\mathbb {R}$ and let $f\in\mathcal{B}(A)$ and $g\in\mathcal{B}_b(A)$ (Borel function on $A$ and bounded Borel function on $A$, resp.) Is it possible to approximate $f$ and/or $g$ by continuous functions on $A$ in a certain sense (pointwise, uniform, etc.)? So yes, how to prove that, or what are the references? So no, what is a counterexample?

Thanks a lot.

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  • $\begingroup$ The uniform limit of continuous functions is continuous. Any pointwise limit of continuous functions is Borel, but not vice versa. In fact, you can stratify the class of Borel functions into $\omega_1=\aleph_1$ many sub-classes, by starting with the continuous functions, continuing with Baire one functions (the pointwise limits of sequences of continuous functions), then with Baire two functions (pointwise limits of Baire one functions), etc. $\endgroup$ – Andrés E. Caicedo Feb 23 '15 at 23:17