On a parametric functional equation of Dhombres type

In this paper we consider the functional equation f (x f (x)) = k f (x)2, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f : R^+ \rightarrow R^+ $$\end{document} and k > 0 is a real parameter. We give a characterization of the class of its continuous solutions, and show that there are discontinuous solutions which are strongly irregular.