On strong quasi-continuity of functions of two variables
Let \(E \in \{\mathbb{R}, \mathbb{R}^2\}\). If \(A \subset E\) and \(x \in E\), then \(d_u (A,x)\) denotes the upper outer density of \(A\) at \(x\). (If \(E = R^2\), then we consider the ordinary differentiation basis in \(\mathbb{R}^2.)\) Let \(f : E \to \mathbb{R}\) and \(x \in E\). We will write \(x \in A(f)\) if there is an open set \(U \subset E\) such that \(f |(U \cup \{x\})\) is continuous at \(x\) and \(d_u (U,x) > 0\). We will write \(x \in B(f)\) if for every \(\varepsilon > 0\) we have \(d_u (U,x) > 0\), where \(U = \text{int} \{t \in E : |f(t) - f(x) |< \varepsilon\}\). We will write \(x \in Q_s (f)\) (and say that \(f\) is strongly quasi-continuous at \(x)\) if for every \(\varepsilon > 0\) and every density open set \(U \ni x\) there is an open set \(V\) such that \(V \cap U \neq \emptyset\) and \(|f(t) - f(x) |< \varepsilon\) for \(t \in V \cap U\). We will denote \(A = \{f : A(f) = E\}\), \(B = \{f : B(f) = E\}\), and \(Q_s = \{f : Q_s (f) = E\}\). In the paper under review the author proves several results concerning sets \(A(f)\), \(B(f)\), and \(Q_s (f)\), as well as the families \(A\), \(B\), and \(Q_s\). Main results: 0. For every function \(f\) we have \(A(f) \subset B(f) \subset Q_s (f)\). 1. If \(E \backslash B(f)\) is a null set, then \(f\) is continuous almost everywhere. 2. \(Q_s = B\). 3. The uniform closure of \(A\) is equal to \(Q_s\).
scientific article; zbMATH DE number 878217