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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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marked as duplicate by José Carlos Santos calculus Sep 25 at 14:02

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  • $\begingroup$ 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. $\endgroup$ – Bib-lost Sep 25 at 13:57
  • $\begingroup$ If $g(x)$ is a discontinuous solution to Cauchy's Functional Equation then $f(x)=e^{g(\ln x)}$ satisfies your functional equation. $\endgroup$ – lulu Sep 25 at 14:01