We study a notion of generalized H\"older continuity for functions on $\mathbb{ R}^d$. We show that for any bounded function $f$ of bounded support and any $r>0$, the $r$-oscillation of $f$ defined as $osc_r f (x):= \sup_{ B_r(x)} f - \inf_{ B_r(x)} f$ is automatically generalized H\"older continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.
Categories: math.MG math.DS