Algebraic properties of the class of Sierpiński-Zygmund functions

We give a condition for a family of functions to be lineable by means of its additivity. The novelty presented here is that we solve the lineability problem using a technique that is not constructive, as are most approaches to this problem. We relate the notions of additivity and lineability and use this relation to give a general method to find the lineability of large families of functions. We also study more examples of pathologically behaving functions, in particular, the class of Jones functions, which is a highly pathological subclass of perfectly everywhere surjective functions. We work on the additivity, lineability, and main properties of this class.

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“…Similarly to what happened with e c , it is also known (see [9,Thm. 2.5]) that d c can take as value any regular cardinal between c + and 2 c .…”

Section: Lineability and Additivitymentioning

confidence: 72%

Lineability and additivity in RR

Gámez

Muñoz-Fernández

Seoane‐Sepúlveda

2010

Journal of Mathematical Analysis and Applications

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“…In [3,Theorem 4.9] it is shown that f ∈ R(R, R) if and only if there is an h ∈ R R such that h • f ∈ SZ.…”

Section: Resultsmentioning

confidence: 99%

“…The two propositions above suggest the following two problems about the cardinal C out (SZ) which are posed in [3]. We first give a combinatorial characterization of C out (SZ) which is a corollary of the two following general theorems.…”

Section: Resultsmentioning

confidence: 99%

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Compositions of Sierpiński-Zygmund Functions and Related Combinatorial Cardinals

Jordan¹

2001

Journal of Applied Analysis

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“…The cardinal function A(F), for F R X , is defined as the smallest cardinality of a family G ⊆ R X for which there is no g ∈ R X such that g + G ⊆ F. It was investigated for many different classes of real functions; see e.g., [4,5,10]. Recall here that A(F) ≥ 3 is equivalent to F − F = R X (see [12,Proposition 1]).…”

Section: Introductionmentioning

confidence: 99%

On functions whose graph is a Hamel basis

Płotka

2002

Proc. Amer. Math. Soc.

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Abstract. We say that a function h : R → R is a Hamel function (h ∈ HF) if h, considered as a subset of R 2 , is a Hamel basis for R 2 . We prove that every function from R into R can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all f 1 , f 2 ∈ R R there is a g ∈ R R such that g + f 1 , g + f 2 ∈ HF. We show that this fails for infinitely many functions.

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Algebraic properties of the class of Sierpiński-Zygmund functions

Cited by 26 publications

References 12 publications

Lineability and additivity in RR

Lineability and additivity in RR

Compositions of Sierpiński-Zygmund Functions and Related Combinatorial Cardinals

On functions whose graph is a Hamel basis

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