Points of continuity of quasiconvex functions on topological vector spaces
Let \({(X,\mathcal{T})}\) be a Baire topological vector space and \({f:X\to\overline{\mathbb{R}}}\) be a quasiconvex function, that is \({f(\lambda x+(1-\lambda)y)\leq\max\{f(x),f(y)\}}\) for all \({x,y\in X}\) and every \({\lambda\in[0,1]}.\) In the article, the author investigates continuity properties of such functions. He gives necessary and sufficient conditions for a quasiconvex function \(f\) to be continuous on the residual subset of~\(X,\) the proof of quasicontinuity for any upper semicontinuous quasiconvex function~\(f,\) and other additional statements. The proofs of all facts base on the consideration of an topological essential extrema NEWLINE\[NEWLINE \mathcal{T}\mathrm{ess\,sup}f=\sup\{\alpha\in\mathbb{R}:\{f\mid_U>\alpha\}\;\text{is of second category}\} NEWLINE\]NEWLINE for an open \({U\subset X},\) and rely on some properties of convex sets. Some examples are considered in the article.
scientific article; zbMATH DE number 6212965