yesterday night, I was studying sequence of functions in $\mathbb R$ and then this question came to mind.
When a sequence of real valued function is given, we can find out it's pointwise limit function on the same domain.
what about the converse?
Suppose a function $f$ on $E\subseteq \mathbb R$ is given. Can we find some sequence of functions $\{f_n\}$ on $E$ such that $\lim f_n=f$ holds for all $x\in E$ ?
If yes, how many such is possible? meaning by, will this sequence of function be unique or there will be plenty of such? If plenty of such , then will it be countable or uncountable?
I tried to figure out but the only intuition that came to mind was "yes, there will be some sequence of functions converging pointwise to $f$". But after that, I am unable to establish it. Moreover, i have no idea for the other consequences