Wikipedia:Reference desk/Archives/Mathematics/2021 February 4

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February 4

Continuously differentiable implies Holder continuous

On 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 $f$ be a function that is continuously differentiable on some closed and bounded interval $I$. Then it has a derivative $f'$ 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 $K$ on the interval, so for all $x\in I$, $|f'(x)|\leq K.$ Lipschitz continuity now follows (with that constant $K$) from the mean value theorem.  --Lambiam 14:33, 4 February 2021 (UTC)