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What are all the solutions to the functional equation $f(ax)=bf(x)$, where $a,b>0$, and $f$ is continuous, strictly monotone and increasing, and $x$ ranges over the reals? references? proof?
Additional details following the first response:
It is easy to see that the function $f(x)=c\cdot x^\alpha$, with $\alpha =\log b/\log a$ and any constant $c$ is a solution for the functional equation. Also, the $c$ can be different for the positives and the negatives. So, if $x^\alpha$ is not monotone itself, we can create a monotone solution by gluing together a positive $c$ for the positive side with a negative $c$ for the negative side. Is this correct?
My question is if these are all the solutions, and if this is appears in the literature.