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Additive Sierpiński-Zygmund functions. (English) Zbl 1110.26002

A function \(f: R \to R\) belongs to the class of Sierpiński-Zygmund functions (SZ for short) if the restriction \(f \setminus A\) is discontinuous for each \(A \subset R\) of size \(c\). The function \(f\) is called additive if \(f(x+y) = f(x) + f(y)\) for every \(x,y \in R\). The authors primarily consider the Gibson’s diagram and show that almost all inclusions from Gibson’s diagram remain strict in the class of all additive Sierpiński-Zygmund functions. For this they construct suitable examples. In particular to show that there exists an additive SZ function which is Darbaux but not connectivity and to show the existence of an additive SZ function which is almost continuous and satisfies Cantor intermediate value property the authors assume that cov\((M) = c\) where cov\((M)\) denotes the minimal cardinality of a family of meager sets which cover \(R\). Also to show the existence of an additive SZ function which is connectivity but not almost continuous they assume the continuum hypothesis. In the last section the authors examine when the inverses of additive one-to-one SZ functions defined on subspaces of \(R\) are also of SZ type. In the two main results they show that under the assumption that cov\((M) = c\), as there exists an additive bijection \(f\) such that both \(f, f^{-1}\) belong to SZ, also there exists an additive bijection which belongs to SZ but its inverse does not belong to SZ.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
03E50 Continuum hypothesis and Martin’s axiom
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