  Mode-locking regions (resonance tongues) formed by border-collision
bifurcations of piecewise-smooth, continuous maps commonly exhibit a
distinctive sausage-like geometry with pinch points called "shrinking points".
In this paper we extend our unfolding of the piecewise-linear case [{\em
Nonlinearity}, 22(5):1123-1144, 2009] to show how shrinking points are
destroyed by nonlinearity. We obtain a codimension-three unfolding of this
shrinking point bifurcation for $N$-dimensional maps. We show that the
destruction of the shrinking points generically occurs by the creation of a
curve of saddle-node bifurcations that smooth one boundary of the sausage,
leaving a kink in the other boundary.
