Regularity properties of functional equations and inequalities
Using \textit{R. Trautner}'s theorem [Q. J. Math. Oxf. Ser. II 38, 127-130 (1987; Zbl 0611.26002)] the author proves the main theorems: 1. Let R and T topological spaces. Suppose that R has a countable base and that X is metrizable. If g: \(X\to R\) and H: \(R\times X\to T\) are mappings were g is measurable on a set M of finite positive Haar measure and H is continuous in its first variable, then any solution f: \(X\to T\) of \(f(x\cdot y)=H(g(x),y)\) for \(x,y\in X\), is continuous. 2. Let G: \(X\times X\to R\) be a mapping. If there is a subset M of X of positive finite Haar measure, such that for each \(y\in X\) the mapping \(x\mapsto G(x,y)\) is bounded above on M, then any solution f: \(X\to R\) of f(x\(\cdot y)\leq G(x,y)\) for \(x,y\in X\), is locally bounded above.
scientific article; zbMATH DE number 4107333
scientific article