On an alternative Cauchy equation in two unknown functions. Some classes of solutions (Q1180496)

The authors consider the alternative Cauchy functional equation (1) \(g(xy)\neq g(x)g(y)\Rightarrow f(xy)=f(x)f(y)\), where \(f\), \(g\) map a topological group \((X,\cdot)\) into a group \((S,\cdot)\). In the first part of the paper they prove the following. Theorem: Let \((S,\cdot)\) be a Hausdorff topological group and \(X(\cdot)\) a connected topological group with the property that there is a fundamental system \(\mathcal U\) of open neighbourhoods of \(e\) in \(X\) such that for all \(U\in{\mathcal U}\) and all ``local'' homomorphisms \(a: U\to S\), i.e. \(a(xy)=a(x)a(y)\) for all \(x,y\in U\) such that \(xy\in U\) there is some \(b\in\hbox{Hom}(X,S)\) such that \(b\mid_ U=a\). Furthermore assume that either \(X\) is locally connected or separable. Finally, let \((Y,\cdot)\) be a topological group which is a continuous homomorphic image of \(X\) and \((f,g)\) a continuous solution of (1) on \(Y\). Then the solution is ``trivial'', i.e. either \(f\) or \(g\) belongs to \(\hbox{Hom}(Y,S)\). After that they investigate a wider class of solutions in the case when \(X=\mathbb{R}^ n\).