Research of differentiable modulo $p^n$ functions

For the class $D_n$ of differentiable modulo $p^n$ functions, subsets $A_n$, $B_n$, $C_n$ are defined so that every function $f$ in $D_n$ is uniquely represented by the sum of certain functions $f_A\in A_n$, $f_B\in B_n$, $f_C\in C_n$. The numbers of functions, of bijective functions and of transitive functions in $D_n$ are found via this representation. According to these cardinality relations, the set of transitive differentiable modulo $p^2$ functions coincide with the set of transitive polynomial functions, but this ceases to be true with increasing the degree of the modulo. It is shown that a function $f$ in $D_n$ is invertible if and only if $f$ is invertible modulo $p$ and the derivatives of $f$ are not equal 0 modulo $p^i$, $i=2,\dots,n$. A recurrent formula is presented for finding inverse differentiable modulo $p^n$ function for a bijective function in $D_n$. A transitivity condition is obtained for a differentiable modulo $p^n$ function. It is shown that any transitive function $f$ in $D_n$ may be constructed from a function $\hat f$ in $D_{n-1}$ such that $f=\hat f\pmod{p^{n-1}}$.