$$f_n(x)=(1+x^n)^{\frac{1}{n}}, x\in[0,2]$$
Show that this sequence is uniformly convergent to a function which's not differentiable at 1.
While trying to show uniform convergence:
When $0\le x \le 1: \lim_{n \rightarrow \infty} f_n(x)=1$.
When $x\gt 1: \lim_{n \rightarrow \infty} f_n(x)=x$.
So, How is $f_n$ uniformly convergent when its pointwise limit is not continuous?