Let G denote the set G = {f : R → R | f is inﬁnitely diﬀerentiable at every point x ∈ R}. (R as in the reals)

(a) Prove that G is a group under addition. Is G a group under multiplication? Why or why not?

(b) Consider the function ϕ : G → G deﬁned by ϕ(f) = f′

Prove that ϕ is a homomorphism with respect to the group operation of addition. What is the kernel of ϕ?

(c) Consider the function ψ : G → G deﬁned by ψ(f) = f′′ − f. Prove that ψ is a homomorphism with respect to the group operation of addition. What is the kernel of ψ?