Find a sequence of differentiable functions $\{f_n\}$ on $[0,1]$ such that $f_n \to 0$ uniformly, but $\{f_n'(1/2)\}$ does not converge to $0$.
I have tried to find an example which follows above condition, but failed to approach. Please help me!
2026-04-13 01:38:15.1776044295
Find a sequence of differentiable functions $\{f_n\}$ on $[0,1]$ such that $f_n \to 0$ uniformly, but $\{f_n'(1/2)\}$ does not converge to $0$.
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Consider, for example, the fucntion $$f_n(x)=\frac{sin(2n\pi x)}{\sqrt n}$$ Clearly $f_n \to 0$ uniformly but $\{f_n'(1/2)\}$ diverges.