Some Additive Darboux–like Functions

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. 1991 Mathematics Subject Classification. Primary 26A15, 26A30; Secondary 03E50.