Commutative ring and integral domain

(a) Let F(R) = all f: R -->R

For f,g belongs to F(R) define f+g: R -->R by x-->f(x)+g(x)

and f.g:R-->R by x-->f(x)g(x)

Show that F(R) is a commutative ring which is not an integral domain

(b) Define C(R), D(R) to be all f in F(R) which are respectively continuous and differentiable. Show that D(R)=<C(R)=<F(R)

Calculate F(R)* and U(F(R))

Note: I used R to denote for Real number not Ring R because I dont use Latex

In part a,I think we have to show somehow F(R) satisfy all the conditions of a ring under + and . ? how about integral domain, what do we have to prove?

In part b, I have no idea, please show me how to do it or give me some hints?

Thanks alot for your time