Two examples concerning almost continuous functions

AbstractIn 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