I'm trying to think of a function which is Lipschitz but which fails to be differentiable at a dense set of points. Of course, being a Lipschitz function it will be differentiable almost everywhere, but can we make it so that the function is not differentiable at every rational point (for instance)?
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In his answer to the linked duplicate question, Dave L. Renfro gives an example of a function which satisfies the requirements of this question. – Xander Henderson May 10 '23 at 19:58