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We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given increasing homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under $\varphi$ and which contains in its interior no fixed point except for $1$. They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. A characterization of the class $𝒮(J,\varphi )$ 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 $\varphi$ is increasing. In the present paper we solve the converse problem, for which continuous maps $f(0,\infty )\to J$, where $J$ is an interval, there is an increasing homeomorphism $\varphi$ of $J$ such that $f\in 𝒮(J,\varphi )$. We...