It is true that if $f\in C^{1,\alpha}(I)$ than $f\in C^{0,1}(I)$? I mean: if $f$ is bounded and differenciable with bounded and holder continuous derivative, then $f$ is bounded and lipschitz continuous? Here $I$ is an interval, NOT necessarily bounded. If it is bounded it is clear.
2026-03-25 21:44:59.1774475099
Is $C^{1,\alpha}\subseteq C^{0,1}$?
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Let $I = \mathbb R$ and let $f(x) = x^2$. Then $f'$ is Lipschitz continuous, but $f$ is not.