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 plain. In Section 3 we will construct, under different additional set theoretical assumptions, a function f: R --> R which is additive, almost continuous, has the strong Cantor intermediate value property, but is not extendable. In Section 4 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.
Mathematics Subject Classification: 26A15, 26A30 (03E50)
Keywords: additive, Darboux, almost continuous, extendability functions
Date received: July 7, 1997.
Date published: July 9, 1997.