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Suppose f(x) has a derivative at x=0, that is f'(0) exists.

Is it necessarily true that f(x) is differentiable in some open interval containing x=0?

Others--who know calculus better than I--say no, f(x) is not necessarily differentiable for x≠0. So my question is, what is an example function where that is the case? That is,

f'(x) exists at x=0

f'(x) does not exist for x

I'm unable to think of an example, but am quite curious about this.f'(x) does not exist for x

*close to*zero