Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road

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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 Sierpinski–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 Sierpinski–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 Sierpinski–Zygmund function with a perfect road at each point.