On the solutions of a functional equation of Dhombres
The authors give a characterization of general solutions of the functional equation \(f(xf(x)) = f(x)^2\), \(x\in R\). They show that there is a solution whose graph has the full outer two-dimensional Lebesgue measure and has the Darboux property. It turns out that partial restrictions, like boundedness, monotonicity and/or continuity on a subinterval do not imply the continuity of solutions. They provide some techniques of constructing the monotonic solutions \(f\) whose points of discontinuity form a dense set in \(R\).
scientific article; zbMATH DE number 785224