Does uniform convergence of continuously differentiable $f_n$ imply point-wise convergence of derivatives?

Question

Let $f_n : [0, 1] \rightarrow \Bbb R$ be a sequence of continuously differentiable functions, if $f_n \rightrightarrows f$. Can we conclude $f_n' \rightarrow f'$ pointwise?

I know there are many counterexamples for "uniform convergence" of derivatives, for example in this question: this

If not, I really need a counterexample.

Are you assuming that $f$ is differentiable? – José Carlos Santos Mar 01 '22 at 02:44 — José Carlos Santos, Mar 01 '22 at 02:44

score 1 · Accepted Answer · answered Mar 01 '22 at 02:49

1

If $f_n(x)=\frac1n\sin(nx)$, then $f_n\rightrightarrows0$. But $f_n'(x)=\cos(nx)$, and therefore $(f_n')_{n\in\Bbb N}$ does not converge pointwise to the null function.

answered Mar 01 '22 at 02:49

José Carlos Santos

427,504

It's a duplicate. Found quickly with a Google search for “pointwise convergence of derivatives site:math.stackexchange.com” – Martin R Mar 02 '22 at 08:03
@MartinR Indeed. But this time I tried a lot to find a duplicate and I was unable to do it. – José Carlos Santos Mar 02 '22 at 08:16

Does uniform convergence of continuously differentiable $f_n$ imply point-wise convergence of derivatives?

1 Answers1