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 Banaszewski (1997). (See also Question 5.5 of Gibson and Natkaniec (1996–97).) 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 strictly increasing. This property is used to construct a ZFC example of an additive almost continuous function f :R→R which has the strong Cantor intermediate value property but is not extendable. This answers a question of Rosen (1997–98). This also generalizes Rosenʹs result (1997–98) that a similar (but not additive) function exists under the assumption of the Continuum Hypothesis, and gives a full answer to Question 3.11 of Gibson and Natkaniec (1996–1997).