  The stable and unstable manifolds of an invariant set of a piecewise-smooth
map are themselves piecewise-smooth. Consequently, as parameters of a
piecewise-smooth map are varied, an invariant set can develop a homoclinic
connection when its stable manifold intersects a non-differentiable point of
its unstable manifold (or vice-versa). This is a codimension-one bifurcation
analogous to a homoclinic tangency of a smooth map, referred to here as a
homoclinic corner. This paper presents an unfolding of generic homoclinic
corners for saddle fixed points of planar piecewise-smooth continuous maps. It
is shown that a sequence of border-collision bifurcations limits to a
homoclinic corner and that all nearby periodic solutions are unstable.
