On weak linear continuous functions

This is what I have first asked in SE but I think it is more suitable for here. I am interested in the set of all continuous functions $f: (0, \infty) \longrightarrow \Bbb{R}$ with the following property:

For any irrational positive number $\alpha$ there exists a real number $b_\alpha$ such that for any $x > 0$ we have $f(\alpha x) = b_\alpha f(x)$.

Any reference or suggestions would be helpful.

• What kind of information would you like? Where did this come up? It will be hard to say anything unless you provide more information. – Matthias Ludewig Jan 22 '14 at 23:00

$f$ must be a power function, $f(x)= cx^e$. Proof:
Assume $f(x) \neq 0$ for some $x$, Then $b_\alpha= f(\alpha x)/f(x)$ for all $\alpha$, hence $b_\alpha$ is a continuous function $(0,\infty)-\mathbb Q^{>0} \to \mathbb R$.
$b_{\alpha \beta} = f(\alpha \beta x)/f(x) = b_\alpha f(\beta x)/f(x)= b_\alpha b_\beta$. Hence $b_\alpha$ is a continuous group homomorphism.
We can adapt the usual proof that any continuous group homomorphism $\mathbb R \to \mathbb R$ is linear to a proof that $b_\alpha$ is a power function $x \to x^c$. Hence $f(x) = c x^e$ for some constant $c$.
• what is the domain of $b_\alpha$? – user40021 Jan 23 '14 at 14:21