For a function defined on $[0,1]\subset \mathbb{R}$, we call it reverse-Holder with exponent $\alpha >0$ if there exists some $c > 0$ such that $$|f(x) - f(y)| \geq c|x-y|^\alpha. $$ The definition can be useful when studying the Hausdorff dimension, for instance see Reverse Hölder Continuity and Hausdorff dimension.
However, while Holder-continuous functions are ubiquitous, reverse-Holder functions seems hard to imagine. Intuitively, reverse Holder condition states that the function has to be highly-irregular/rough everywhere in the domain. How does one construct such a function? Can it still be continuous or even Holder continuous?
P.S. When I think of rough functions, I think of things like sample path of Brownian motion and Weierstrass function. However, I believe none of them are reverse holder.