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I've seen the following theorem:

If $(f_n)_n$ is a sequence of $C^1$ functions in $[a,b]$, such that $(f_n(c))_n$ converges to $f(c)$ for some $c\in [a,b]$, and $(f_n')_n$ converges uniformly to $g$. Then $(f_n)_n$ converges uniformly to f such that $f'=g$.

The question I have is if the convergence of $f$ for some $c$ is really necessary? I can't think of an example in which $(f_n)_n$ is a sequence of $C^1$ functions in $[a,b]$, such that $(f_n')_n$ converges uniformly to $g$, but $(f_n(c))_n$ is not convergent for all $c\in [a,b]$

Any ideas?

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    $\begingroup$ $f_n(x) = x+n$. $\endgroup$ – Daniel Schepler Sep 22 '17 at 18:03
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    $\begingroup$ How about $f_n = n$ $\endgroup$ – Coolwater Sep 22 '17 at 18:04
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    $\begingroup$ This came up just minutes ago. $\endgroup$ – mechanodroid Sep 22 '17 at 18:08
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    $\begingroup$ $f_n = (-1)^n$. $\endgroup$ – PhoemueX Sep 22 '17 at 19:44

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