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On some subclasses of the family of Darboux Baire 1 functions

Ivanova G., Wagner-Bojakowska E.

Opuscula Mathematica

2014

Vol. 34, no. 4

777--788

We introduce a subclass of the family of Darboux Baire 1 functions f : R → R modifying the Darboux property analogously as it was done by Z. Grande in [On a subclass of the family of Darboux functions, Colloq. Math. 17 (2009), 95–104], and replacing approximate continuity with I-approximate continuity, i.e. continuity with respect to the I-density topology. We prove that the family of all Darboux quasi-continuous functions from the first Baire class is a strongly porous set in the space DB1 of Darboux Baire 1 functions, equipped with the supremum metric.

The Darboux property for polynomials in Golomb's and Kirch's topologies

Szczuka P.

Demonstratio Mathematica

2013

Vol. 46, nr 2

429--435

In this paper, we present conditions which are equivalent to the Darboux property for non-constant polynomials in Golomb's and Kirch's topologies on the set of positive integers.

On sets of some classes and their tengency

Konik T.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2003

Vol. 2, nr 1

61--68

In this paper the problem of the tangency of sets of some classes having the Darboux property in a generalized metric space is considered. Some sufficient and necessary conditions for the tangency of sets of these classes have been given here.

On the sets of continuity points of Darboux quasicontinuous functions

Grande M.

Demonstratio Mathematica

2003

Vol. 36, nr 4

819--825

We show that for every dense G-set A C R there is a Baire 1 Darboux bilaterally quasicontinuous function f : R-R such that C(f) = A, where C(f) denotes the set of all continuity points of f. Moreover we prove that for each G-set A there is a Baire 2 Darboux function f : R-R such that C(f) = A.

On the products of bounded Darboux Baire one functions

Maliszewski A.

Journal of Applied Analysis

1999

Vol. 5, nr 2

171-185

It is shown that for each k > 1, if f is a Baire one function and f is the product of k bounded Darboux (quasi-continuous) functions, then f is the product of k bounded Darboux (quasi-continuous) Baire one functions as well.