Abstract
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 $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph. This answers a question of Gibson [15]. The second example is that of a connectivity function $F\colon\mathbb{R}^2\to\mathbb{R}$ 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 $f\colon\mathbb{R}\to\mathbb{R}$ with dense graph such that $f^{-1}(0)$ is countable. We also give a sufficient condition for a Darboux function $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable.
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