The problem I am struggling with is as follows:

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


where a>0 and 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 f^{-1} is the inverse function of f.

Thank you for any help.