Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a bounded measurable function.
Is it possible to find a sequence of functions $\{f_n \}_n: \mathbb{R} \to \mathbb{R}$ in $C^{\infty}_c( \mathbb{R})$ such that $f_n(x) \uparrow f(x)$, for every $x \in \mathbb{R}$?
This seems to be true without the assumption that the convergence is from below, by standard results from the literature. However, I am really not sure about this case.
Best Answer
No. It can be shown that a pointwise limit of a sequence of continuous functions is continuous on a dense set. [This is a consequence of Baire Category Theorem]. However, a bounded measurable function can be discontinuous at every point.