  We show that if the graph of a bounded analytic function in the unit disk
$\mathbb D$ is not complete pluripolar in $\mathbb C^2$ then the projection of
the closure of its pluripolar hull contains a fine neighborhood of a point $p
\in \partial \mathbb D$. On the other hand we show that if an analytic function
$f$ in $\mathbb D$ extends to a function $\mathcal{F}$ which is defined on a
fine neighborhood of a point $p \in \partial \mathbb D$ and is finely analytic
at $p$ then the pluripolar hull of the graph of $f$ contains the graph of
$\mathcal{F}$ over a smaller fine neighborhood of $p$. We give several examples
of functions with this property of fine analytic continuation. As a corollary
we obtain new classes of analytic functions in the disk which have non-trivial
pluripolar hulls, among them $C^\infty$ functions on the closed unit disk which
are nowhere analytically extendible and have infinitely-sheeted pluripolar
hulls. Previous examples of functions with non-trivial pluripolar hull of the
graph have fine analytic continuation.
