Question: Suppose that f : (0, ) R satisfies f(x) - f(y) = f(x/y) for all x, y (0, ) and f(l) = 0.

Suppose that f : (0, ∞) → R satisfies f(x) - f(y) = f(x/y) for all x, y ∊ (0, ∞) and f(l) = 0.
a) Prove that f is continuous on (0, ∞) if and only if f is continuous at l.
b) Prove that f is differentiable on (0, ∞) if and only if f is differentiable at 1.
c) Prove that if f is differentiable at 1, then f'(x) = f'(l)/x for all x ∊ (0, ∞).
[If f'(l) = l, then f(x) = log x.]

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