  When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and
common vertex at the origin) our main result provides a sufficient condition
for any map $L$, that is continuous and piecewise linear relatively to this
slicing, to be invertible. Some examples show that the assumptions of the
theorem cannot be relaxed too much. In particular, convexity of the slices
cannot be dropped altogether when $n=4$. This result cannot be plainly extended
to a greater number of slices. Our result is proved by a combination of linear
algebra and topological arguments.
