Mainly, is this true and why/why not? I am trying to apply the intermediate value theorem to f' but I'm not entirely sure if it's true and I can't find any definitive answers.
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4See Discontinuous derivative. for how bad the derivative of a function can be if it is differentiable. The intermediate value property holds for $f'$, though. – Sarvesh Ravichandran Iyer Apr 10 '23 at 03:58