Let $f:(0,\infty)\to \Bbb R$ be a continuous function and $\alpha,\beta>0$ be two fixed real numbers.
Under what conditions on $\alpha$ and $\beta$ can we deduce that $f$ is a constant function if we know $$ f(x)=f(\alpha x) = f(\beta x)? $$
I noticed that if $\alpha$ is an integer power of $\beta$ then the statement is not true. Indeed, for $\alpha=4,\beta=2$ the function $$ f(x) = \sin(2\pi \log_2(x)) $$ provides a counterexample.
I believe that if $\frac{\log\alpha}{\log\beta}\in \Bbb R\backslash\Bbb Q$ then the statement should hold. For example, I think it is likely that $f(x)=f(2x)=f(3x)$ implies $f$ is constant but I don't know how to prove it.
Writing $g(x)=f(e^{x})$ we can reduce the problem to the following: if $a$ and $b$ (non-zero) are periods of a continuous function $g$ under what conditions can we say $g$ is a constant. The answer is $a/b$ is irrational. Hence the answer to the present question is $f$ is a constant if $\frac {\log \, \alpha} {\log\, \beta}$ is irrational.