Research of differentiable modulo p functions

For the class of differentiable modulo p functions, subsets , C _n are defined so that every function f in is uniquely represented by the sum of certain functions f _A G A _n, f _B G , f _C G C _n. The numbers of functions, of bijective functions and of transitive functions in are found via this representation. According to these cardinality relations, the set of transitive differentiable modulo p 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 is invertible if and only if f is invertible modulo p and the derivatives of f are not equal 0 modulo p , i = 2, . . . , n. A recurrent formula is presented for finding inverse differentiable modulo p function for a bijective function in . A transitivity condition is obtained for a differentiable modulo p function. It is shown that any transitive function f in may be constructed from a function f in D _n-1 such that f = f (mod p ).

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