prove $f(x)f(y)=f(xy), f(1)=1 \iff f(x)=x^k (k\ real)$ for $f:\mathbb{R}^+\to \mathbb{R}^+$
I find $f(a^r)=f(a)^r$ for rational r, but I cannot move to the next step.
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hskimse
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Do we additionally assume that $f$ is continuous? Otherwise the statement seems false to me. If you have continuity, then you can use a density argument. – Bib-lost Sep 25 '19 at 13:57
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If $g(x)$ is a discontinuous solution to Cauchy's Functional Equation then $f(x)=e^{g(\ln x)}$ satisfies your functional equation. – lulu Sep 25 '19 at 14:01