Sur la quasi-continuité et la quasi-continuité approximative. (On quasi-continuity and approximate quasi-continuity). (French) Zbl 0657.26003

Let (R,\({\mathcal T})\) be a topological space of the set of all reals. A function \(f:R\to R\) is called \({\mathcal T}\)-quasi-continuous (\({\mathcal T}\)- cliquish) iff for every point \(x\in R,\) every \({\mathcal T}\)-open set U containing x and every \(\epsilon >0\) there exists a \({\mathcal T}\)-open set V such that \(V\subset U\) and \(| f(t)-f(x)| <\epsilon\) for each \(t\in V\) \((_{V} f<\epsilon).\) Let \({\mathcal T}_ e({\mathcal T}_ d)\) be the Euclidean (density) topology on R and Q, \(Q_ d\), resp. (P, \(P_ d\), resp.) be the set of all \({\mathcal T}_ e\)-quasi-continuous, resp. \({\mathcal T}_ d\)-quasicontinuous (\({\mathcal T}_ e\)-cliquish, resp. \({\mathcal T}_ d\)-cliquish) real functions of a real variable. If K is any family of real functions of a real variable, then we put \(B(K)=\{f:R\to R:\) there exists a sequence \(\{f_ n\}^{\infty}_{n=1}\) of K such that \(f(x)=\lim_{n\to \infty}f_ n(x)\) for each \(x\in R\}.\) Let D be the set of all \(f:R\to R\) with the Baire property. There is proved: \(B(Q)=P,\) \(B(P)=D,\) \(B(Q_ d)=P_ d\) and there exists a topology \({\mathcal T}\) on R such that \(B(Q_{{\mathcal T}})\neq P_{{\mathcal T}},\) where \(Q_{{\mathcal T}}(P_{{\mathcal T}})\) is the set of all \({\mathcal T}\)-quasi-continuous (\({\mathcal T}\)-cliquish) functions.
Reviewer: L.Mišík


26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
Full Text: DOI EuDML