So, I have a somewhat arbitrary question about this topic but first I need to state the theorem I am familiar with and from which I will base my question.
Theorem: Let $(f_n)_n:[a,b] \to \mathbb{R}$ be $C^1$ functions. If: $$f_n {\rightrightarrows f} \quad \text{and} \quad f_n ' {\rightrightarrows g} $$
Then f is differentiable and also $f'=g$
Now with that in mind, my question is:
Supose I define $(f_n)_n$ and $f \enspace$in compact sets, as above, still with $f_n {\rightrightarrows f}$ , but this time $f_n $ are smooth functions;
Also, suppose I don't even know if $f_n'$ converges uniformly onto something.
When/what hypothesis would be necessary to ensure that $f_n'\to f'$?
where the single arrow denotes pointwise convergence.