Two inductive constructions of weird real functions

This is a joint work with Krzysztof Ciesielski. We construct, under the assumption that cov(M)=c, an additive connectivity function f from reals to reals which is not almost continuous. This answers a question of D.Banaszewski. We also give a ZFC example of an almost continuous function f from reals to reals which has the strong Cantor intermediate value property but is not extendable. This generalizes a result of H.Rosen, who constructed such a function under the assumption of CH.