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