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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.

On weakly measurable functions

Żeberski S.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

421-428

We show that if T is an uncountable Polish space, X is a metrizable space and f : T → X is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f [T ∖ M] is a separable space. We also give an example showing that “metrizable” cannot be replaced by “normal”.

On cliquishness of maps of two variables

Ewart J.

Demonstratio Mathematica

2002

Vol. 35, nr 3

657-670

We give some sufficient conditions under which a separately cliquish map of two variables is cliquish.

Category analogue of sup-measurability problem

Ciesielski K., Shelah S.

Journal of Applied Analysis

2000

Vol. 6, nr 2

159-172

A function F: R2→ R is called sup-measurable if Ff : R→ R given by Ff(x) = F(x,f(x)), x ∈R, is measurable for each measurable function f: R→ R. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analogues. In this paper we will show that the existence of the category analogues of sup-measurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is the subject of a work in prepartion by Rosłanowski and Shelah.

On the Peano derivaties of functions having the Baire property

Hejduk J.

Demonstratio Mathematica

1998

Vol. 31, nr 3

663-668

We present the result that the set of points, where the I-approximate Peano derivatives of function having the Baire property exists, has the Baire property.