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.