  The present paper develops two concepts of pointwise differentiability of
higher order for arbitrary subsets of Euclidean space defined by comparing
their distance functions to those of smooth submanifolds. Results include that
differentials are Borel functions, higher order rectifiability of the set of
differentiability points, and a Rademacher result. One concept is characterised
by a limit procedure involving inhomogeneously dilated sets.
  The original motivation to formulate the concepts stems from studying the
support of stationary integral varifolds. In particular, strong pointwise
differentiability of every positive integer order is shown at almost all points
of the intersection of the support with a given plane.
