Sequence of functions is uniformly convergent to a function (f). Is sequence of their derivatives converging to a derivative (f') at only one point?

Question

For each $n \in N$ let the function $f_n : \mathbb{R} \to \mathbb{R}$ be differentiable, let the function $f : \mathbb{R} \to \mathbb{R}$ be differentiable and let $f_n \rightrightarrows f$. Is it possible that the sequence $f'_{n}(x)$ converges to $f'(x)$ for exactly one $x ∈ R$? If it is give an example.

Answer 1

Consider $f_{n}(x)=n^{-1/2}\cos(n^{3/2}x)$. While $f_{n}(x)$ uniformly converges to $0$, $f_{n}’(x)$ converges to $0$ only when $x=0$.