Let $[a,b]\subset\mathbb R$ be some compact non-degenerate interval. We know that if a sequence of differentiable functions $f_n:[a,b]\to\mathbb R$ converges pointwise in at least one point $c\in[a,b]$ and $f_n'$ converges uniformly to a function $g:[a,b]\to\mathbb R$ then $f_n\to f$ uniformly where $f'=g$.
Now I ask about the hypothesis of $f_n(x)$ being convergent in at least one point of $[a,b]$: Is there an example of a sequence of differentiable functions $f_n:[a,b]\to\mathbb R$ such that the sequence of derivatives $(f_n')$ converge uniformly to $g$ but $(f_n(x))$ doesn't converge for any point $x\in[a,b]$?
This is easy to verify if the sequence $f_n$ is not bounded, as in $f_n(x)=n$, then $f_n'\to0$ uniformly but $(f_n(x))$ doesn't converge on any point.
So the actual question is: Is there an example of a bounded sequence (there's $A>0$ such that $|f_n(x)|<A$ for all $n\in\mathbb N,x\in[a,b]$) of differentiable functions $f_n:[a,b]\to\mathbb R$ such that the sequence of derivatives $(f_n')$ converge uniformly to $g$ but $(f_n(x))$ doesn't converges for any point $x\in[a,b]$? And what conditions we could add so that we don't need the hypothesis of $(f_n(x))$ converging for some point $x\in[a,b]$ to conclude that $f'=g$?
You can use the same idea but replace $n$ by a bounded sequence that diverges, and that doesn't depend on $x$. For example, $f_n(x) = (-1)^n$. Here $f_n' = 0$ converges uniformly toward $0$ and $f_n$ diverges at each point.