Sequence of uniform convergent sequence $\{f_n\}$ such that $\{f_n'\}$ does not converge

Question

I would like to find uniformly convergent sequence of differentiable functions $f_n:[0,1]\to\mathbb{R}$ such that the sequence $f_1'$, $f_2'$,$f_3'\ldots$ does not converge.

Answer 1

For $n\in\mathbb{N}$ let $f_n : [0,1] \to \mathbb{R}$ be defined as $$f_n(t) = \frac{1}{n}\sin(n^2t) ,\quad t \in [0,1]$$

$(f_n)_{n=1}^\infty$ converges uniformly to $0$:

$$\|f_n\|_{\infty} = \sup_{t\in[0,1]} \frac{1}{n}\left|\sin(n^2t)\right| \le \frac1n \xrightarrow{n\to\infty} 0$$

The derivatives are given by $$f'_n(t) = n\cos(n^2t), \quad t \in [0,1]$$ The sequence $(f'_n)_{n=1}^\infty$ does not converge since $f_n'(0) = n$, which is unbounded.