On the solutions of a functional equation of Dhombres
type: Item
instance of: scholarly article
title: On the solutions of a functional equation of Dhombres
zbMATH Open document ID: 0860.39030
DOI: 10.1007/BF03322840
published in: Results in Mathematics
publication date: 1997-04-13T00:00:00Z
review text: 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 \(R\).
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 785224
zbMATH Keywords: transfinite sequence
zbMATH Keywords: outer Lebesgue measure
zbMATH Keywords: Dhombres functional equation
zbMATH Keywords: general solutions
zbMATH Keywords: Darboux property
zbMATH Keywords: monotonic solutions
author: Peter Kahlig
author: Jaroslav Smítal
reviewed by: Marek Cezary Zdun
MaRDI profile type: Publication
cites work: Q3854084
full work available at URL: https://doi.org/10.1007/bf03322840
OpenAlex ID: W2044864837
description: scientific article; zbMATH DE number 785224