Smooth compositions with a nonsmooth inner function

Let p:R→R be a given function, and let Ap be the set of smooth functions f such that f(p(⋅)+c) is smooth for any c∈R. We show that if p is not smooth, then either every element of Ap is constant, or there is a nonzero constant d such that Ap is equal to the set of smooth functions of periodicity d.