8. Suppose that f : R ---+ R satisfies f(x + y) = f(x) + f(y) for each x, y E R.

(a) Show that f(nx) = nf(x) for all x E Rand n E Z.

(b) Prove that f(qx) = qf(x) for all x E Rand q E Q.

(c) Prove that f is continuous at 0 if and only if f is continuous on R.

(d) Prove that if f is continuous at 0, then there is an mER such that f(x) =

mx for all x E R.

[!]. This exercise is used in Section 7.4.