Does uniform convergence of $f$ imply convergence of derivatives?

Question

Let $X$ denote the collection of all differentiable functions $f : [0, 1] \rightarrow \Bbb R$, such that $f(0)=0$ and $f'$ is continuous.

Let $\{f_n\}$ be a Cauchy sequence. By Cauchy criterion for uniform convergence, $f_n$ converges uniformly to some $f$.

Does that imply that $f'_n \rightarrow f'$ uniformly?

Answer 1

No! Imagine the $f_n$ getting close to $f$ uniformly, but getting bumpier and bumpier. You should be able to use this idea to come up with a counterexample.