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I've not been able to give a example of a sequence of functions that converges uniformly in $\mathbb{R}$ such that the sequence of the derivatives does not converge punctually in any point of $\mathbb{R}$.

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    Something like $(1/\sqrt n)\sin(nx)$. – Sergei Golovan Jun 04 '17 at 19:51
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    Consider infinitely many upper semicircles of radius $1$, joint together at their endpoints, like the letter "m". Let this be the graph of $f_1$. Now draw a similar graph, but make the radii $1/2$ instead of $1$. This will be the graph of $f_2$. Go on like this. These functions will converge uniformly to $f =0$. – ThePortakal Jun 04 '17 at 20:00
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    I second portakal's notion, and recall a particular example that I saw while perusing the forum a few months back

    https://math.stackexchange.com/questions/2227191/uniform-convergence-and-lengths

     – M A Pelto Jun 04 '17 at 22:46

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