F y.

8. Suppose F F is differentiable ∀x > 0 ∀ x > 0 and F(xy) = F(x) + F(y) F ( x y) = F ( x) + F ( y), ∀x, y > 0 ∀ x, y > 0 . Prove that if F F is not the zero function, then ∃ ∃ a > 0 a > 0 such that: F(x) =loga(x) F ( x) = log a. ⁡. ( x), ∀x > 0 ∀ x > 0 . I seem to be doing fine except on getting the base a a for the log.

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With F − n = ( − 1)n + 1Fn, the recurrence holds for all integers, positive and negative. And of course Fn = ϕn + ( − ϕ) − n √5. We assume f is continuous and bijective. And satisfies f(f(x)) = f(x) + x Proved in another answer: f(0) = 0 . So f is either increasing or decreasing. Let's consider the increasing case.

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