Problem

Suppose that f : (0,∞) → R satisfies f (x) − f (y) = f (x/y) for all x, y ∈ (0,∞) and f (1...

Suppose that f : (0,∞) → R satisfies f (x) − f (y) = f (x/y) for all x, y ∈ (0,∞) and f (1) = 0.

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


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 ′(1)/x for all x ∈ (0,∞).

[Note: If f ′(1) = 1, then f (x) = log x.]

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Solutions For Problems in Chapter 4.1