Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous

Ciesielski,; Miller, '

doi:10.2307/44152549

Cited by 21 publications

(14 citation statements)

References 9 publications

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“…(a) c + ≤ A(AC) = A(Conn) = A(D) = A(PES) = e c ≤ 2 c and this is all that can be proved in ZFC, see [14,24] or [13,Proposition 1.8]. (b) A(Ext) = A(PR) = c + , see [17].…”

Section: Additivity Coefficient: Definition and Backgroundmentioning

confidence: 92%

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Additivity Coefficients for all Classes in the Algebra of Darboux-Like Maps on $$\mathbb {R}$$

et al. 2020

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The class $$\mathbb D$$ D of generalized continuous functions on $$\mathbb {R}$$ R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property. The algebra $$\mathcal {A}(\mathbb D)$$ A ( D ) of classes of functions generated by these families contains 17 atoms. In this work we will calculate the values of the additivity coefficient $${{\,\mathrm{A}\,}}(\mathcal {F})$$ A ( F ) for all atoms $$\mathcal {F}$$ F in the algebra $$\mathcal {A}(\mathbb D)$$ A ( D ) . We also determine the values $${{\,\mathrm{A}\,}}(\mathcal {F})$$ A ( F ) for a lot of other families $$\mathcal {F}\in \mathcal {A}(\mathbb D)$$ F ∈ A ( D ) . Open questions and new directions of research shall also be provided.

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“…(a) c + ≤ A(AC) = A(Conn) = A(D) = A(PES) = e c ≤ 2 c and this is all that can be proved in ZFC, see [14,24] or [13,Proposition 1.8]. (b) A(Ext) = A(PR) = c + , see [17].…”

Section: Additivity Coefficient: Definition and Backgroundmentioning

confidence: 92%

“…see [14], and let PES stand for the family of all perfectly everywhere surjective maps f ∈ R R , that is, such that f [P ] = R for every perfect set P ⊂ R. Also, following [35], we define…”

Section: Additivity Coefficient: Definition and Backgroundmentioning

confidence: 99%

Additivity Coefficients for all Classes in the Algebra of Darboux-Like Maps on $$\mathbb {R}$$

et al. 2020

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“…Then f + g ∈ F for g = h − f . To see (3) note that for F = R R and every g ∈ R R there is f ∈ F with f + g ∈ F, namely f = h − g, where h ∈ R R \ F. Proof. (1) is obvious.…”

Section: A(f) = Min{|fmentioning

confidence: 99%

“…The functions A and M for the classes AC, Conn and D were studied in [10,3,12]. In particular, the following is known.…”

Section: A(f) = Min{|fmentioning

confidence: 99%

Cardinal Invariants Concerning Extendable and Peripherally Continuous Functions

Ciesielski¹

1995

Real Analysis Exchange

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Let F be a family of real functions, F ⊆ R R. In the paper we will examine the following question. For which families F ⊆ R R does there exist g : R → R such that f + g ∈ F for all f ∈ F ? More precisely, we will study a cardinal function A(F) defined as the smallest cardinality of a family F ⊆ R R for which there is no such g. We will prove that A(Ext) = A(PR) = c + and A(PC) = 2 c , where Ext, PR and PC stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from R into R, respectively. In particular, the equation A(Ext) = c + immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson [6]. We will also study the multiplicative analogue M(F) of the function A(F) and we prove that M(Ext) = M(PR) = 2 and A(PC) = c. This article is a continuation of papers [10, 3, 12] in which functions A(F) and M(F) has been studied for the classes of almost continuous, connectivity and Darboux functions.

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“…The following theorem is proved in [5]. Using a similar technique as in the case of the above theorem, we will prove the following proposition.…”

Section: Generalized Continuity Versus Additivity 41 Introductionmentioning

confidence: 83%

Set-theoretic and algebraic properties of certain families of real functions

Płotka¹

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Set-theoretic and Algebraic Properties of Certain Families of Real Functions Krzysztof P lotka Given two families of real functions F 1 and F 2 we consider the following question: can every real function f be represented as f = f 1 + f 2 , where f 1 and f 2 belong to F 1 and F 2 , respectively? This question leads to the definition of the cardinal function Add: Add(F 1 , F 2) is the smallest cardinality of a family F of functions for which there is no function g in F 1 such that g + F is contained in F 2. This work is devoted entirely to the study of the function Add for different pairs of families of real functions. We focus on the classes that are related to the additive properties and generalized continuity. Chapter 2 deals with the classes related to the generalized continuity. In particular, we show that Martin's Axiom (MA) implies Add(D,SZ) is infinite and Add(SZ,D) equals to the cardinality of the set of all real numbers. SZ and D denote the families of Sierpiński-Zygmund and Darboux functions, respectively. As a corollary we obtain that the proposition: every function from R into R can be represented as a sum of Sierpiński-Zygmund and Darboux functions is independent of ZFC axioms. Chapter 3 is devoted entirely to the classes related to the concept of additivity. We introduce the definition of Hamel functions. We say that a real function is a Hamel function if its graph is a Hamel basis for the plane. Main result of this chapter is the theorem that every real function can be represented as the pointwise sum of two Hamel functions. In Chapter 4 we investigate the function Add for pairs of classes such that one relates to the generalized continuity and the other to the additive properties.

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Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous

Cited by 21 publications

References 9 publications

Additivity Coefficients for all Classes in the Algebra of Darboux-Like Maps on $$\mathbb {R}$$

Additivity Coefficients for all Classes in the Algebra of Darboux-Like Maps on $$\mathbb {R}$$

Cardinal Invariants Concerning Extendable and Peripherally Continuous Functions

Set-theoretic and algebraic properties of certain families of real functions

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