On some modification of Darboux property

Abstract

We introduce some family of functions

Comparison of Some Subfamilies of Functions Having The Baire Property

“…In particular, we have this property for the classes of functions connected with some modifications of Darboux property (see [2], [3], [11]). As, according to the last theorem, DQ is strongly porous in D and each strongly porous set is nowhere dense, DQ is nowhere dense in D.…”

“…For this purpose, we need some auxiliary lemmas. All of them are similar to the lemmas from [5]. However, in this paper, we consider other families of functions, so the proofs of these lemmas need some essential modifications.…”

On some Modification of Świtąkowski Property

“…Let B be a right-hand interval set at 0 such that 0 is a right-hand I-density point of B (see [6]). Obviously, f B ∈ D τ I \D s .…”

“…Let C be a right-hand interval set at 0 such that 0 is a right-hand I-dispersion point of C. Then, f C is not I-approximately continuous at 0 (see [6] for more details) and assumes value 1 only at point 0, so f C ∈ DQ\D τ I . Let us introduce a metric ρ in the space UQ in the following way:…”

“…Let I be a σ-ideal of the sets of the first category. In [6] and [7], we introduce a family of functions f : R → R modifying the Darboux property analogously as it was done by Z. Grande and replacing approximate continuity with I-approximate continuity, i.e., the continuity with respect to I-density topology in the domain (see [3], [19], [20], [23], [24]).…”