Considering in particular the set in Rn where the v-directional derivative of u fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of Rademacher's theorem.
Hence, for any countable collection of unit vectors v1, v2, ..., there is a single set E of measure zero such that the gradient and each vi-directional derivative exist everywhere on the complement of E, and are linked by the dot product.
[4] Rademacher's theorem can be used to prove that, for any p ≥ 1, the Sobolev space W1,p(Ω) is preserved under a bi-Lipschitz transformation of the domain, with the chain rule holding in its standard form.
[6] Rademacher's theorem is also significant in the study of geometric measure theory and rectifiable sets, as it allows the analysis of first-order differential geometry, specifically tangent planes and normal vectors.
[7] Higher-order concepts such as curvature remain more subtle, since their usual definitions require more differentiability than is achieved by the Rademacher theorem.