Title: A note on surjectivity of piecewise affine mappings,
Abstract: A standard theorem in nonsmooth analysis states that a piecewise affine
function $F:\mathbb R^n\rightarrow\mathbb R^n$ is surjective if it is
coherently oriented in that the linear parts of its selection functions all
have the same nonzero determinant sign. In this note we prove that surjectivity
already follows from coherent orientation of the selection functions which are
active on the unbounded sets of a polyhedral subdivision of the domain
corresponding to $F$. A side bonus of the argumentation is a short proof of the
classical statement that an injective piecewise affine function is coherently
oriented.