TITLE: Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$?
QUESTION [9 upvotes]: Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$?
This seems to be true,but I haven't found such a conclusion in any textbook.

REPLY [20 votes]: For every bounded Lipschitz-function, this is true. To see this, note that for $|x-y| \leq 1$, we have
$$
|f(x) - f(x)| \leq C \cdot |x-y| = C\cdot |x-y|^\alpha \cdot |x-y|^{1-\alpha} \leq C\cdot |x-y|^\alpha
$$
and for $|x-y| > 1$, we have
$$
|f(x) - f(y)| \leq|f(x)| + |f(y)| \leq 2K \leq 2K \cdot |x-y|^\alpha.
$$
As any Lipschitz function on a bounded domain is bounded, the claim holds in particular for Lipschitz functions on bounded domains.
But for unbounded functions, this is not true in general. For example, $f : \Bbb{R} \to \Bbb{R}, x\mapsto x$ is trivially Lipschitz, but
$$
\frac{|f(n) - f(0)|}{|n-0|^\alpha} = n^{1-\alpha} \to \infty \text{ for } n \to \infty,
$$
so that $f$ is not $\alpha$ Hölder-continuous for any $\alpha \in (0,1)$.