Suppose that f : R R satisfies f(x + y) = f(x) + f(y) for each

Question:

Suppose that f : R → R satisfies f(x + y) = f(x) + f(y) for each x, y ∊ R.
a) Show that f(nx) = nf(x) for all x ∊ R and n ∊ Z.
b) Prove that f(qx) = qf(x) for all x e R and q ∊ 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 m ∊ R such that f(x) = mx for all x ∊ R.