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 Canter 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 tb and every perfect set K between g(a) and g(b) then is a perfect set C subset of (a, b) such that g[C] subset of K and g up arrow 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 Canter intermediate value property but is not extendable. This answers a question of Rosen (1997-98). This also generalizes Rosen's 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). (C) 2000 Elsevier Science B.V. All rights reserved.