On sets of discrete convergence points of sequences of real functions. (English) Zbl 1070.26005

Wesołowska, Jolanta

Real Anal. Exch. 29(2003-2004), No. 1, 107-120 (2004).

For a sequence \((a_n)\) of real numbers, \(a\in\mathbb{R}\) is said to be the discrete limit of \((a_n)\) iff there exists \(k\in\mathbb{N}\) such that \(a_n= a\) for \(k< n\) [cf. Á. Császár and M. Laczkovic, Studia Sci. Math. Hung. 10, 463–472 (1975; Zbl 0405.26006)]. The purpose of the paper is to characterize, for different families \({\mathcal F}\) of functions \(f: \mathbb{R}\to\mathbb{R}\), the set \(L^d({\mathcal F})\) of points \(x\in\mathbb{R}\), where given a sequence \((f_n)\subset{\mathcal F}\), \(f_n(x)\) discretely converges to some limit \(f(x)\). As \({\mathcal F}\), Baire class \(\alpha\), Darboux functions, measurable functions, derivatives, approximately continuous functions, quasi-continuous functions, etc. are considered.

Reviewer: Ákos Császár (Budapest)

MSC:

26A21	Classification of real functions; Baire classification of sets and functions
26A03	Foundations: limits and generalizations, elementary topology of the line
54C50	Topology of special sets defined by functions

Keywords:

sequence of functions; sets of convergence points; discrete convergence

Citations:

Zbl 0405.26006