If function f is continuously differentiable at some point, say x=0, and is Lipschitz in some neighborhood of x=0, is that true there is an open neighborhood of x=0 in which f is continuously differentiable?
I know there is a function which is differentiable at just one point and continuous everywhere else. I also know the set of continuity of a derivative of a function is dense. I also familiar with a differentiable function which is not C1 On the Cantor set.
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