The pointwise behavior of Sobolev-type functions, whose weakderivatives up to a given order belong to somerearrangement-invariant Banach function space, is investigated. Weintroduce a notion of approximate Taylor expansion in norm forthese functions, which extends the usual definition of Taylorexpansion in Lp-sense for standard Sobolev functions. Anapproximate Taylor expansion for functions in arbitrary-orderSobolev-type spaces, with sharp norm, is established. As aconsequence, a characterization of those Sobolev-type spaces inwhich all functions admit a classical Taylor expansion is derived.In particular, this provides a higher-order versionof a well-known result of Stein [27] on thedifferentiability of weakly differentiable functions. Applicationsof our results to customary classes of Sobolev-type spaces are alsopresented.
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