On boundary behavior of arbitrary functions defined in a half-space
Let \({\mathbb{R}}^ N_+=\{x=(x_ 1,x_ 2,...,x_ N):\) \(x_ N>0\}\) and let \(\Pi^ N=\partial {\mathbb{R}}^ N_+\). If E is a subset of \(\Pi^ N\), a point \(\zeta\) in the closure of E is called a point of porosity of E if there exist a real number \(\epsilon >0\), a sequence of points \(\{a_ n\}\) in \(\Pi^ N\) converging to \(\zeta\), and a sequence of N-1 dimensional balls \(\{B_ n\}\) contained in \(\Pi^ N\), where \(B_ n\) has center at \(a_ n\) and radius \(r_ n\) and \(B_ n\cap E=\emptyset\), such that \(r_ n/\rho_ n>\epsilon\), where \(\rho_ n\) is the distance from \(a_ n\) to \(\zeta\). Let f be a function defined on \({\mathbb{R}}^ N_+\) which takes its values in a locally compact Hausdorff space. A point \(\zeta \in \Pi^ N\) is called a VV-singular point for f if there exist two disjoint open cones \(V_{\zeta}\) and \(V'_{\zeta}\) in \({\mathbb{R}}^ N_+\), each having its vertex at \(\zeta\), such that the cluster sets of f at \(\zeta\) relative to \(V_{\zeta}\) and relative to \(V'_{\zeta}\) are not equal. The author proves that, for such a function f, the set E of VV-singular points for f has the form \(E=\cup p(F_ n)\), where \(\{F_ n\}\) is a sequence of closed subsets of \(\Pi^ N\) and \(p(F_ n)\) denotes the set of all points of porosity of the set \(F_ n\). It is further shown that if E is a subset of \(\Pi^ N\) of this form, then there exists a function g defined on \({\mathbb{R}}^ N_+\), where g is non-negative, continuous, and bounded, such that E is exactly the set of VV-singular points for g.
scientific article; zbMATH DE number 4163226
scientific article