Two examples concerning almost continuous functions

In this note we will construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f : R → R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski [2]. (See also [12, Question 5.5].) We will also show that every extendable function g : R → R with a dense graph satisfies the following stronger version of the SCIVP property: for every a < b and every perfect set K between g(a) and g(b) there is a perfect set C ⊂ (a, b) such that g[C] ⊂ K and g ↾ C is continuous ∗AMS classification numbers: Primary 26A15, 26A30; Secondary 03E50.