We consider the functional equation f(xf(x)) = ϕ(f(x)) where ϕ: J → J is a given homeomorphism of an open interval J ⊂ (0, ∞) and f : (0, ∞) → J is an unknown continuous function. A characterization of the class S(J,ϕ) of continuous solutions f is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f : (0, ∞) → J, where J is an interval, there is an increasing homeomorphism ϕ of J such that f ∈ S(J,ϕ). We also show why the similar problem for decreasing ϕ is difficult.