A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is
$$ f(x) = \left( \frac{\beta}{\alpha^{\log\_2 \beta}} \right) \alpha^{\log\_2 x}. $$


Do we know whether or not the solution is unique?


**Edit:** the function should be real-valued of a real variable, continuous, and monotonic.



Uniqueness of solutions of functional equations
No, it is not unique. To obtain the general solution, set $g(t)=\log f(e^t)$.
Then your equation is equivalent to $g(t+\log 2)=g(t)+\log\alpha$. One solution is an 
affine function, but any periodic function can be added to it. So there are infinitely many solutions and continuity and monotonicity do not help.


