In this paper we show that if the real line R is not a union of less than continuum many 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:R-->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. (U. B. Darji, A Sierpinski-Zygmund function which has a perfect road at each point, Colloq. Math. 64 (1993), 159--162.)
The paper contains also a construction (in ZFC) of an additive Sierpinski-Zygmund function with perfect road at each point.
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