  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.
