While uniform convergence preserves continuity, it does not preserve differentiability. Find an explicit example of a sequence of differentiable function on $[-1,1]$ that converge uniformly to a function $f$ such that $f$ is not differentiable. Hint: Consider $|x|^{1+\frac 1n}$, show that these functions are differentiable, converge uniformly, and then show that the limit is not differentiable.
Following Hint, let $f_n(x)=|x|^{1+\frac 1n}$.
Let $c \in [-1,1].$ Then, $$\lim_{x \to c} \frac {|x||x|^{1/n}-|c||c|^{1/n}}{x-c}=?.$$
Show that $f_n(x)$ converges uniformly to $f(x).$ I claim that $f(x)=|x|.$ Note that $|x|^{1/n}\le 1 $ for all $x\in [-1,1]$. Then, $$\lim_{n \to \infty}||f_n-f||_u=\lim_{n \to \infty}\sup(|x||x|^{1/n}-|x|)\le\lim_{n \to \infty}\sup(|x|-|x|)=\lim_{n \to \infty}\sup(0)=0.$$
And we know that $|x|$ is not differentiable at $0$.
Could you tell me how to prove $f_n$ is differentiable at $[-1,1]$?