Two examples concerning extendable and almost continuous functions

by

Krzysztof Ciesielski & Harvey Rosen

Real Anal. Exchange 25(2) (1999--2000), 579-598.

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function f:[0,1]-->[0,1] with a G\delta graph. This answers a question of Gibson. The second example is that of a connectivity function F:R2-->R with dense graph such that F-1(0) is contained in a countable union of straight lines. This easily implies the existence of an extendable function f:R-->R with dense graph such that f-1(0) is countable.

We also give a sufficient condition for a Darboux function f:[0,1]-->[0,1] with a G\delta graph whose closure is bilaterally dense in itself to be quasicontinuous and extendable.