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.

Exposition of some parts (included Abstract and Introduction) revised. Proof of Lemma 5.2 slightly modified to correct a mistake. Some references added