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At the top of the wikipedia articles on Hölder condition, Lipschitz Continuity and others, the chain of logic is as follows: On a compact subspace of a metric space: $$Continuously Differentiable \subseteq Lipschitz Continuous \subseteq \alpha-Hölder Continuous \subseteq Uniformly Continuous \subseteqq Continuous$$

There's a nice proof $Uniformly Continuous \subseteqq Continuous$ on compact metric spaces here.

$f(x) = x^2$ is a function which is continuous, but not uniformly continuous. It is however continuously differentiable, but not Lipschitz continuous.

My question is, do any of the inclusions above hold when we relax the imposition of compactness? (is, for example, the branch Lipschitz $\subseteq$ Hölder always true?)

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  • Well, LipschitzContinuous $⊆$ $α$-HölderContinuous $⊆$ UniformlyContinuous $⊆$ Continuous holds in general. Compactness gives you ContinuouslyDifferentiable $⊆$ LipschitzContinuous and Continuous $⊆$ UniformlyContinuous (note the order). – Adam Bartoš Oct 29 '14 at 17:06
  • (I mean $α$-Hölder for $a ∈ (0, 1)$, for $α=1$ we got Lipschitz and for $α=0$, we got bounded. Note that continuous $⊆$ bounded on compact.) – Adam Bartoš Oct 29 '14 at 17:10

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