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.
Full text on line in pdf format. Requires Adobe Acrobat Reader.
Requires rae.cls file amsmath.sty, amssymb.sty, and epic.sty.
Last modified October 13, 2000.