On a parametric functional equation of Dhombres type (Q1271581)

Consider the functional equation (1): \( f( x f (x)) = k f(x)^2 \), where \( f: R^+ {\o}R^+\) and \( k > 0 \). For \( k \leq 0 \) there are only trivial solutions. The case \( k = 1\) was studied by \textit{J. Dhombres} [C. R. Acad. Sci., Paris, Sér. A 281, 809-812 (1975; Zbl 0344.39009)] finding the continuous solutions and discontinuous solutions were found by \textit{P. Kahlig} and \textit{J. Smital} [Result. Math. 27, No. 3-4, 362-367 (1995; Zbl 0860.39030)]. In the present paper the authors characterize the class of continuous solutions of (1) for \( k \neq 1 \) and show with some clever arguments that there are discontinuous solutions which are strongly irregular.