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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 Gδ graph. This answers a question of Gibson [15]. 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 Gδ graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable.

TWO EXAMPLES CONCERNING EXTENDABLE AND ALMOST CONTINUOUS FUNCTIONS