General Pexider equations. II: An application of the theory of webs (Q1087755)

\textit{H. W. Pexider} [Monatsh. Math. Phys. 14, 293-301 (1903)] generalized the Cauchy functional equation in studying \(h(x+y)=f(x)+g(y)\). In the first of these papers the author investigates \(h\circ T=F\circ \pi\), where T is a given continuous function from an open connected set \(\Omega\) of \({\mathbb{R}}^ n\) to \({\mathbb{R}}\), F is a given function, strictly monotonic in each variable separately, mapping an open connected set \({\tilde \Omega}\) of \({\mathbb{R}}^ n\) to \({\mathbb{R}}\), h is an unknown function from T(\(\Omega)\) to \({\mathbb{R}}\), and \(\pi\) is an unknown product mapping from \(\Omega\) to \({\tilde \Omega}\). He proves that, if T is one- to-one in each variable, then any continuous solution \(\pi\) must be injective or constant on \(\Omega\), and, conversely, that if an injective solution \(\pi\) exists then T must be one-to-one in each variable. In the second paper he uses the web structure of \textit{W. Blaschke} and \textit{G. Bol} [Geometrie der Gewebe (1938; Zbl 0020.06701)] to prove that, if in addition to the above-mentioned hypotheses, T is strictly monotonic, then any continuous solution \(\pi\) is uniquely determined on \(\Omega\) by its values at two points of \(\Omega\) ; if a solution \(\pi\) is not continuous on \(\Omega\), then \(\pi\) (\(\omega)\) is dense in \({\tilde \Omega}\) for every open \(\omega\) in \(\Omega\) ; and if a solution \(\pi\) is continuous at one point of \(\Omega\) it is continuous on \(\Omega\).