In this note we will construct several additive Darboux-like functions ƒ : ℝ → ℝ answering some problems from (an earlier version of) [Gibson, Natkaniec, Real Anal. Exchange, 22: 492–533, 1996–97]. In particular, in Section 2 we will construct, under different additional set theoretical assumptions, additive almost continuous (in sense of Stallings) functions ƒ : ℝ → ℝ whose graph is either meager or null in the plane. In Section 3 we will construct an additive almost continuous function ƒ : ℝ → ℝ which has the Cantor intermediate value property but is discontinuous on any perfect set. In particular, such an ƒ does not have the strong Cantor intermediate value property.
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