My friend asks me about this problem.
$f$ is differentiable on $(a,b)$, for $c\in (a,b)$, exists a sequence $x_n\to c$ and $\lim f'(x_n)=f'(c)$, how to show that $f'$ doesn't have to be continuous at $c$?
I know 'there exists' such a sequence does not guarantee that it's true for any sequence converging to $c$, but I can't think of a counterexample. Can anybody prove it or find a counterexample? This problem is kinda interesting and I want to know the answer:) Thanks!!!