  We prove that nonlocal minimal graphs in the plane exhibit generically
stickiness effects and boundary discontinuities. More precisely, we show that
if a nonlocal minimal graph in a slab is continuous up to the boundary, then
arbitrarily small perturbations of the far-away data produce boundary
discontinuities.
  Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a
small perturbation of the prescribed conditions produces boundary
discontinuities.
  The proof relies on a sliding method combined with a fine boundary regularity
analysis, based on a discontinuity/smoothness alternative. Namely, we establish
that nonlocal minimal graphs are either discontinuous at the boundary or their
derivative is H\"older continuous up to the boundary. In this spirit, we prove
that the boundary regularity of nonlocal minimal graphs in the plane "jumps"
from discontinuous to $C^{1,\gamma}$, with no intermediate possibilities
allowed.
  In particular, we deduce that the nonlocal curvature equation is always
satisfied up to the boundary.
  As an interesting byproduct of our analysis, one obtains a detailed
understanding of the "switch" between the regime of continuous (and hence
differentiable) nonlocal minimal graphs to that of discontinuous (and hence
with differentiable inverse) ones.
