Let p be a prime. Consider the following statement.
Any mapping from the residue class ring Z/pZ to itself can be represented as a polynomial function.
True, in fact it is not even necessary that p is a prime.
That is incorrect.
True.
Try to represent mod 4 the function f given by
f(0) = 1, f(1) = 0, f(2) = 0, f(3) = 0.
Consider the
polynomial map x ->
x(x-2)(x-3)···(x-p+1).
It is always zero unless x = 1. Can you use that?
Correct.