On $p$-adic functions preserving Haar measure

Let $\{a_n\}_{n=0}^\infty$ be a uniformly distributed sequence of $p$-adic integers. In the present paper we study continuous functions close to differentiable ones (with respect to the $p$-adic metric); for these functions, either the sequence $\{f(a_n)\}_{n=0}^\infty$ is uniformly distributed over the ring of $p$-adic integers or, for all sufficiently large $k$, the sequences $\{f_k(\varphi_k(a_n))\}_{n=0}^\infty$ are uniformly distributed over the residue class ring $\operatorname{mod}p^k$, where $\varphi_k$ is the canonical epimorphism of the ring of $p$-adic integers to the residue class ring $\operatorname{mod}p^k$ and $f_k$ is the function induced by $f$ on the residue class ring $\operatorname{mod}p^k$ (i.e., $f_k(x)=f(\varphi_k(x))(\operatorname{mod}p^k)$). For instance, these functions can be used to construct generators of pseudorandom numbers.