  The main goal of this paper is to develop a concept of approximate
differentiability of higher order for subsets of the Euclidean space that
allows to characterize higher order rectifiable sets, extending somehow well
known facts for functions. We emphasize that for every subset $ A $ of the
Euclidean space and for every integer $ k \geq 2 $ we introduce the approximate
differential of order $ k $ of $ A $ and we prove it is a Borel map whose
domain is a (possibly empty) Borel set. This concept could be helpful to deal
with higher order rectifiable sets in applications.
