Convergence of sequences of functions having some generalized Pawlak properties. (English) Zbl 1119.26006

Grande, Zbigniew

Real Anal. Exch. 30(2004-2005), No. 2, 581-588 (2005).

This paper concerns convergence results for functions \(f:\mathbb R\rightarrow\mathbb R\) satisfying one of the properties: \({\mathcal M_1}\), if the restriction of \(f\) to the set of its points of discontinuities is monotone, and \({\mathcal M_2}\), if the restriction of \(f\) to the set of its points of “approximate” discontinuities is monotone. The author observes that the elements of \({\mathcal M_1}\) are of Baire class 1 and the elements of \({\mathcal M_2}\) are measurable in Lebesgue’s sense and proves that both classes \({\mathcal M_1}\) and \({\mathcal M_2}\) are uniformly closed. Among the main results, the author establishes conditions for a function \(f:\mathbb R\rightarrow\mathbb R\) to be the pointwise limit of a sequence in \({\mathcal M_1}\) and for a measurable function \(f:\mathbb R\rightarrow\mathbb R\) to be the limit of a sequence in \({\mathcal M_2}\). Additional results are given as well as interesting examples.

Reviewer: Marcia Federson (São Paulo)

MSC:

26A15

Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

Measurability; monotone functions; density topology; uniform convergence; pointwise convergence