Is there a function $f$ such that for all $x$ (on a bounded domain):
- $f(x) \in [a,b]$
- $r \times f(x) = f(rx)$
If it makes things easier, we can assume that $\forall x: x \geq 0$. Though a more general solution would be nice.
P.S. I figure that it may be impossible to find such a scaling under some circumstances. For example, a scaling that maps $x_{max}$ to $a$ and $x_{min}$ to $b$ obviously needs $\frac{a}{b}=\frac{x_{max}}{x_{min}}$