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Suppose that $f_k: (0,1) \rightarrow \mathbb{R}$ is a sequence of differentiable functions that converges uniformly to a function $f: (0,1) \rightarrow \mathbb{R}$. Must $f$ be differentiable?

So I'm pretty sure this isn't true but struggling to find a simple counterexample.

SS'
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  • Counter-example here: https://math.stackexchange.com/a/424297/42969. – Martin R Dec 12 '18 at 21:28
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    Easy on $(-1,1)$ with a picture: smooth bowl shaped functions converging to the absolute value function. You can just round the graph of that function with small circular arcs near the origin. The domain is clearly irrelevant to the essential part of the argument. – Ethan Bolker Dec 12 '18 at 21:31
  • For the counterexample linked, $f_n(x)=\sqrt{(x-1/2)^2+1/n}$, which converges uniformly to non-differentiable function $f(x)=|x-1/2|$. Would $f_n(x)=\sqrt{(x)^2+1/n}$ work converging uniformly to $f(x)=|x|$ which we know isn't differentiable at 0? – SS' Dec 14 '18 at 22:27

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