0
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.