The problem I am struggling with is as follows:

Find all continuous, increasing and defined on the whole line R functions $\displaystyle f$ such that the following condition is satisfied:

$\displaystyle af^{-1}(pf(x))=f^{-1}(pf(ax))$,

where $\displaystyle a>0$ and $\displaystyle p \in (0,1)$ are parameters. My guess is that only linear functions satisfy this condition (imprortant is the assumption, that functions are defined on the whole line)

Obviously $\displaystyle f^{-1}$ is the inverse function of $\displaystyle f$.

Thank you for any help.