Thinking in Problems by Alexander A. Roytvarf

Author:Alexander A. Roytvarf
Language: eng
Format: epub
Publisher: Birkhäuser Boston, Boston

(The same equation is satisfied with log a | t |.) Furthermore, a short computation 2f(−t) = f(t 2) = 2f(t) shows that the extension from ℝ+* to ℝ* of the solution f(t) may be only f(|t|); on the other hand, f(|t|) will satisfy (**). Lastly, we must prove that all continuous nonzero solutions of (**) defined on ℝ+* are logarithms. For the continuous solutions, f(e t ) are continuous homomorphisms ℝ,+ → ℝ,+, that is, linear functions over the reals: f(e t ) = ct (c ∈ ℝ). (Why?) For c ≠ 0 we have, denoting k = c −1 and applying the statement from section P10.9* ,

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