Let () is a set of all function from to . Define addition and multiplication in () as follows: > For all f,g

Let ℱ(ℝ) is a set of all function from ℝ to ℝ. Define addition and multiplication in ℱ(ℝ) as follows:

> For all f,g ∈ ℱ(ℝ), (f+g) : ℝ → ℝ is a function defined with :
(f+g)(x) = f(x) + g(x) for all x ∈ ℝ
> For all f,g ∈ ℱ(ℝ), (fg) : ℝ → ℝ is a function defined with :
(fg)(x) = f(x)g(x) for all x ∈ ℝ

a. Prove that ℱ(ℝ) is Abelian group under the addition
b. Is ℱ(ℝ) is abelian group under multiplication
c. Find one element in ℱ(ℝ) that does not have a multiplicative inverse in ℱ(ℝ). Explain how this shows that ℱ(ℝ) is not a group under multiplication.
d. Determine the necessary and sufficient conditions so that the element in ℱ(ℝ) is a unit in ℱ(ℝ).