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?

  • $\begingroup$ 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. $\endgroup$
    – Jose27
    Sep 14 at 21:27
  • $\begingroup$ @Jose27 Oh right, that's like a special case of dominated convergence? $\endgroup$ Sep 14 at 21:30
  • $\begingroup$ 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. $\endgroup$
    – Jose27
    Sep 14 at 21:33
  • $\begingroup$ @Jose27 So this could mean it is likely there is a counterexample obtained from a sequence not satisfying the criteria for dominated convergence. $\endgroup$ Sep 14 at 21:35
  • $\begingroup$ It also looks like this should be useful, then we can reduce to taking integrals over compact intervals: math.stackexchange.com/questions/3100416/… $\endgroup$ Sep 14 at 21:46

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