Arch. Math. Logic (1997) 37: 29–35 c  Springer-Verlag 1997 Sierpi ´ nski-Zygmund functions that are Darboux, almost continuous, or have a perfect road Marek Balcerzak 1 , Krzysztof Ciesielski 2 , Tomasz Natkaniec 3 1 Institute of Mathematics, L ´ ´ od´ z Technical University, Al. Politechniki 11, 90-924 L ´ ´ od´ z, Poland (e-mail: mbalce@krysia.uni.lodz.pl) 2 Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA (e-mail: kcies@wvnvms.wvnet.edu) 3 Department of Mathematics, Pedagogical University, Chodkiewicza 30, 85-064 Bydgoszcz, Poland (e-mail: wspb11@vm.cc.uni.torun.pl) Received February 28, 1996 Abstract. In this paper we show that if the real line R is not a union of less than continuum many of its meager subsets then there exists an almost continuous Sierpi´ nski–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 Sierpi´ nski–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 Sierpi´ nski–Zygmund function with a perfect road at each point. 1 Introduction Our terminology is standard. In particular, the symbols N, Z, Q and R stand for the sets of all: positive integers, integers, rationals and reals, respectively. We shall consider only real-valued functions of one real variable. No distinction is made between a function and its graph. The family of all functions from a set X into Y will be denoted by Y X . The symbol card (X ) will stand for the cardinality of a set X . The cardinality of R is denoted by c. If A is a planar set, we denote its x -projection by dom (A). For f ,g ∈ R R the notation [f = g] means the set {x ∈ R: f (x )= g(x )}. If J is an ideal of subsets of R, then cov (J ) = min{card (F ): F ⊂ J &  F = R} non (J ) = min{card (A): A ⊂ R & A ∈ J }. (See [5].) The ideal of all meager subsets of R is denoted by K . Recall also the following definitions. Mathematics Subject Classification: Primary: 26A15; Secondary: 03E50, 03E65 Correspondence to : M. Balcerzak