We study a notion of generalized Hölder continuity for functions on 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 Hölder continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.