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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