Can $f_n\to f$ uniformly, $f'_n\to g$ uniformly, but $f$ not being differentiable?

Question

Just the question in the title,

I know that if $f_n$ are differentiable, $f_n\to f$ uniformly, $f'_n\to g$ uniformly and $f$ is differentiable, then $f'=g$, so I'm looking for a counterexample if we remove that hypothesis.

Answer 1

Actually the following stronger result holds:

If $f_n$ are differentiable functions on some open set $U$, $f_n\to f$ pointwise on $U$, and $f'_n\to g$ uniformly on $U$, then $f$ is differentiable on $U$ and $f'=g$ on $U$.

So there is no counterexample.