# Smooth sequence of functions converging pointwise to a smooth function and limit of derivatives

In the wikipedia page of uniform convergence, it says that given a sequence $$\{ f_n \}$$ of differentiable real functions (say, over the reals) with the property that it converges pointwise to some function $$f$$, the limit of $$\{f_{n}' \}$$ need not be equal to $$f'$$.

It then gives an example where $$\{ f_n \}$$ converges uniformly to a differentiable $$f$$, but $$\{f_n '\}$$ does not converge even pointwise.

My question is, what if we assume that each $$f_n$$ and their limit $$f$$ are, say, $$C^\infty$$, and $$\{ f_n '\}$$ converges pointwise to some $$C^\infty$$ function $$g$$ as well. Is it now enough to show that $$f' = g$$? Or do we further need to assume uniform convergence? Is there a classic counterexample to this question as well?

• Intersting question! The answer is yes if you know also that $|f_n'|\leq M$ for some constant independent of $n$. I'm not sure right now about the general case. Sep 14 at 21:27
• @Jose27 Oh right, that's like a special case of dominated convergence? Sep 14 at 21:30
• Yep, that's right. Of course, you can get away with a weaker condition like $|f_n'|\leq h$ for some integrable function $h$, but it's still not enough to give the general case unfortunately. Sep 14 at 21:33
• @Jose27 So this could mean it is likely there is a counterexample obtained from a sequence not satisfying the criteria for dominated convergence. Sep 14 at 21:35
• It also looks like this should be useful, then we can reduce to taking integrals over compact intervals: math.stackexchange.com/questions/3100416/… Sep 14 at 21:46