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:**R**^{2}-->**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.

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**Last modified October 13, 2000.**