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

= 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)