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.**