Wikipedia:Reference desk/Archives/Mathematics/2021 February 4
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February 4
editContinuously differentiable implies Holder continuous
editOn Holder condition it is mentioned that continuously differentiable implies Holder continuous. Where can the proof be found? ThanksAbdul Muhsy (talk) 13:46, 4 February 2021 (UTC)
- Note that this only holds (in general) on a closed and bounded non-trivial interval of the real line. Since Lipschitz continuous means the same as 1-Hölder continuous, all we need to prove is that continuous differentiability implies Lipschitz continuity. Let be a function that is continuously differentiable on some closed and bounded interval . Then it has a derivative that is continuous on that interval, and so is the absolute value of the derivative. By the extreme value theorem, the latter attains some maximum on the interval, so for all , Lipschitz continuity now follows (with that constant ) from the mean value theorem. --Lambiam 14:33, 4 February 2021 (UTC)