  In this paper, we derive a necessary condition for a best approximation by
piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain
this result, which is also a necessary and sufficient condition for
inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a
special property of the knots. Then, using this property, we construct a
characterization theorem for best free knots polynomial spline approximation,
which is stronger than the existing characterisation results when only
continuity is required.
