Abstract.
In this paper we show that if the real line \({\Bbb R}\) is not a union of less than continuum many of its meager subsets then there exists an almost continuous Sierpiński–Zygmund function having a perfect road at each point. We also prove that it is consistent with ZFC that every Darboux function \(f\colon{\Bbb R}\to{\Bbb R}\) is continuous on some set of cardinality continuum. In particular, both these results imply that the existence of a Sierpiński–Zygmund function which is either Darboux or almost continuous is independent of ZFC axioms. This gives a complete solution of a problem of Darji [4]. The paper contains also a construction (in ZFC) of an additive Sierpiński–Zygmund function with a perfect road at each point.
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Received February 28, 1996
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Balcerzak, M., Ciesielski, K. & Natkaniec, T. Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road. Arch Math Logic 37, 29–35 (1997). https://doi.org/10.1007/s001530050080
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DOI: https://doi.org/10.1007/s001530050080