Classical and approximate Taylor expansions of weakly differentiable functions

The pointwise behavior of Sobolev-type functions, whose weak derivatives up to a given order belong to some rearrangement-invariant Banach function space, is investigated. We introduce a notion of approximate Taylor expansion in norm for these functions, which extends the usual definition of Taylor expansion in L^p-sense for standard Sobolev functions. An approximate Taylor expansion for functions in arbitrary-order Sobolev -type spaces, with sharp norm, is established. As a consequence, a characterization of those Sobol ev-type spaces in which all functions admit a classical Taylor expansion is derived. In particular, this provides a higher-order version of a well-known result of Stein on the differentiability of weakly differentiable functions. Applica- tions of our results to customary classes of Sobolev-type sp aces are also presented.