5. Suppose that 1 : (0,00) -+ R satisfies I(x) - I(y) = I(x/y) for all x, y E (0,00)

and 1(1) = o.

(a) Prove that 1 is continuous on (0,00) if and only if 1 is continuous at 1.

(b) Prove that 1 is differentiable on (0,00) if and only if 1 is differentiable at 1.

(c) Prove that if 1 is differentiable at 1, then f'(x) = f'(l)/x for all x E (0,00).

(Note: If 1'(1) = 1, then I(x) = log x.)

[!]. This exercise is used in Section 4.2.