We show that, for a class of moduli functions $\omega(\delta)$, $0\leq\delta\leq 2$, the property $|\varphi(\xi)-\varphi(\eta)|\leq\omega(|\xi-\eta|)$, $\xi,\eta\in S^{n-1}$ implies the corresponding property $|u(x)-u(y)|\leq C\omega(|x-y|)$, $x,y\in B^n$, for $u=P[\varphi]$, provided $u$ is a quasiregular mapping. Our class of moduli functions includes $\omega(\delta)=\delta^{\alpha}$ ($0<\alpha\leq1$), so our result generalizes earlier results on H\"older continuity (see [1]) and Lipschitz continuity (see [2]).