On Universally Bad Darboux Functions

Kirchheim,; Natkaniec,

doi:10.2307/44153725

Cited by 7 publications

(3 citation statements)

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“…Let us look closer to the sensitivity of Darboux functions. Kirchheim and Natkaniec [9] proved that if the union of fewer then c of the first category subsets of R is a set of the first category (add(M) = c), then there exists a Darboux function F such that F + f is not Darboux for any continuous and nowhere constant function f . Based on the general method one can prove that the family of linearly sensitive functions with respect to the Darboux property is c-lineable.…”

Section: Lineability Of Linearly Sensitive Functionsmentioning

confidence: 99%

Lineability of Linearly Sensitive Functions

2020

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In the paper we will focus on lineability of some subsets of R [0,1] which are called linearly sensitive. A function f is called linearly sensitive with respect to the property (or condition) (P ) if f has the property (P ) and for any a = 0 the function f + a · id does not have the property (P ). We discuss some general method of proving c-lineability and use this method to examine lineability of the family of all continuous functions linearly sensitive to the Luzin (N )-property, the family of functions linearly sensitive to theŚwiątkowski condition and the family of functions linearly sensitive to the strongŚwiątkowski condition.Mathematics Subject Classification. 46B87, 15A03, 26A15.

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Section: Lineability Of Linearly Sensitive Functionsmentioning

confidence: 99%

Lineability of Linearly Sensitive Functions

2020

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“…(Kirchheim, Natkaniec [91]). If union of less than c many meager subsets of R is meager (thus under CH or MA) then there exists a Darboux function g : R → R such that f + g is not Darboux for every continuous nowhere constant function f :…”

Section: Theorem 37 (mentioning

confidence: 99%

Set Theoretic Real Analysis

Ciesielski¹

1997

Journal of Applied Analysis

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“…Such a function f is called a universally bad Darboux function for F. Determining for which families F the condition c(F) is fulfilled is a problem considered by several authors (see, e.g., [6], [8], [1], [3] and [4]). In particular, if the additivity of the ideal of all first category subsets of R is equal to 2 ω (e.g., if Martin's Axiom or CH hold), then c(C * ) holds for the family C * of all nowhere constant, continuous functions [3]. On the other hand, there is a model of set theory in which c(C * ) fails to hold.…”

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confidence: 99%