Some Darboux-like functions


Krzysztof Ciesielski

J. Appl. Anal. 4(1) (1998), 43-51.

In this note we will construct several additive Darboux-like functions f:R-->R answering some problems from (an earlier version of) a survey Darboux like functions by R.G.Gibson and T.Natkaniec. In particular, in Section 2 we will construct, under different additional set theoretical assumptions, additive almost continuous (in sense of Stallings) functions f:R-->R whose graph is either meager or null in the plane. In Section 3 we will construct an additive almost continuous function f:R-->R which has the Cantor intermediate value property but is discontinuous on any perfect set. In particular, such an f does not have the strong Cantor intermediate value property.

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Last modified January 28, 2007.