We consider continuous solutions f : R+ ! R+ = (0,1) of the functional equation f(xf(x)) = o(f(x)) where o is a given continuous map R+ ! R+. If o is an increasing homeomorphism the solutions are completely described, if not there are only partial results. In this paper we bring some necessary conditions upon a possible range Rf . In particular, if ojRf has no periodic points except for xed points then there are at most two xed points in Rf , and all possible types of Rf and all possible types of behavior of f can be described. The paper contains techniques which essentially simplify the description of the class of all solutions.