Automorphisms of the semigroup of all differentiable functions

TLDR: In this paper, it was shown that the semigroup of all functions mapping R into R that are (finitely) differentiable at each point of R is a semigroup with associative composition.

Abstract: Let R denote the space of real numbers and let D(R) denote the family of all functions mapping R into R that are (finitely) differentiable at each point of R. Since the composition f o g of two differentiable functions is also differentiable and since the composition operation is associative, it follows that D(R) is a semigroup with this operation. Such semigroups have been studied previously. Nadler, in [4], has shown that the semigroup of al differentiable functions mapping the closed unit interval into itself has no idempotent elements other than the identity function and the constant functions. The proof of that result carries over easily to the semigroup D(R).