  Piecewise-smooth maps are used as discrete-time models of dynamical systems
whose evolution is governed by different equations under different conditions
(e.g.~switched control systems). By assigning a symbol to each region of phase
space where the map is smooth, any period-$p$ solution of the map can be
associated to an itinerary of $p$ symbols. As parameters of the map are varied,
changes to this itinerary occur at border-collision bifurcations (BCBs) where
one point of the periodic solution collides with a region boundary. It is well
known that BCBs conform broadly to two cases: {\em persistence}, where the
symbolic itinerary of a periodic solution changes by one symbol, and a {\em
nonsmooth-fold}, where two solutions differing by one symbol collide and
annihilate. This paper derives new properties of periodic solutions of
piecewise-linear continuous maps on $\mathbb{R}^n$ to show that under mild
conditions BCBs of mode-locked solutions on invariant circles must be
nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps
exhibit a sausage-string structure whereby changes to symbolic itineraries
occur at codimension-two pinch points instead of codimension-one
persistence-type BCBs. But the main result is based on the combinatorical
properties of the itineraries, so the impossibility of persistence-type BCBs
also holds when the periodic solution is unstable or there is no invariant
circle.
