On path continuity. (English) Zbl 1106.26004

Marciniak, Mariola

Real Anal. Exch. 29(2003-2004), No. 1, 247-255 (2004).

If \(E\) is a system of paths, then the function \(f\) is \(E\)-continuous at \(x\) if there exists a path \(A\in E(x)\) such that \(f\) restricted to \(A\) is continuous at \(x\). A function \(f\) is called \(E\)-continuous if it is \(E\)-continuous at each \(x\). In the first part of the paper it is proved that if \(X\) is a connected metric space having at least two points, then there exists a system \(E\) of paths such that the family of bounded continuous real functions defined on \(X\) is superporous in the space of all \(E\)-continuous bounded functions. The second part of the paper is devoted to the study of sets of points of continuity of \(E\)-continuous functions defined on a (complete) metric space. There are conditions in terms of local size of paths which assure that the set of poins of continuity is residual.

Reviewer: Władysław Wilczyński (Łódź)

MSC:

26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21	Classification of real functions; Baire classification of sets and functions
26A03	Foundations: limits and generalizations, elementary topology of the line