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. (See Question 5.5 of [GN]: Gibson, Natkaniec, Darboux like functions, Real Anal. Exchange 22 (1996--97), 492-533.) 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 less than b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] is a subset of K and the restriction g|G of g to C is continuous strictly increasing. This property is used to construct a ZFC example of an almost continuous function f:R-->R which has the strong Cantor intermediate value property but is not extendable. This answers a question of H. Rosen. This also generalizes Rosen's result that a similar (but not additive) function exists under the assumption of the continuum hypothesis, and gives a full answer to Question 3.11 from [GN].
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Last modified April 6, 2000.