  Differently from their classical counterpart, nonlocal minimal surfaces are
known to present boundary discontinuities, by sticking at the boundary of
smooth domains.
  It has been observed numerically by J. P. Borthagaray, W. Li, and R. H.
Nochetto ``that stickiness is larger near the concave portions of the boundary
than near the convex ones, and that it is absent in the corners of the
square'', leading to the conjecture ``that there is a relation between the
amount of stickiness on $\partial\Omega$ and the nonlocal mean curvature of
$\partial\Omega$''.
  In this paper, we give a positive answer to this conjecture, by showing that
the nonlocal minimal surfaces are continuous at convex corners of the domain
boundary and discontinuous at concave corners.
  More generally, we show that boundary continuity for nonlocal minimal
surfaces holds true at all points in which the domain is not better than
$C^{1,s}$, with the singularity pointing outward, while, as pointed out by a
concrete example, discontinuities may occur at all point in which the domain
possesses an interior touching set of class $C^{1,\alpha}$ with $\alpha>s$.
