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Some additive Darboux-like functions

Ciesielski K.

Journal of Applied Analysis

1998

Vol. 4, nr 1

43-51

In this note we will construct several additive Darboux-like functions f: R −> R answering some problems from (an earlier version of) [4]. 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.