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?