Gâteaux differentiability of Lipschitz functions via directional derivatives.
Suppose that a Banach space \(X\) is written as the direct sum \(X= V \oplus W\), with dim \(W = n\), where \(0\leq n\leq \)dim \(X\) is an integer. A Lipschitz surface of codimension \(n\) is a subset \(S\) of \(X\) of the form \(S= \{v+h(v) : v\in V\}\) for a Lipschitz function \(h:V\to W,\) a notion introduced by the second author [Commentat. Math. Univ. Carol. 19, 179--189 (1978; Zbl 0404.47025)]. For a Lipschitz function \(f:G\to Y\), defined on an open subset \(G\) of \(X\) and with values in another Banach space \(Y\), denote by \(S_{+}(f)\) (respectively by \(S(f)\)) the set of all points \(x\in G\) for which the one-sided directional derivative \(f'_{+}(x,u)\) (respectively the two-sided directional derivative \(f'(x,u)\)) exists for every \(u\in X\). The authors show that if \(X\) is a separable Banach space, then for every Lipschitz function \(f:G\to Y\) the set \(S_{+}(f)\) can be covered by a countable union of Lipschitz surfaces of codimension 1. The same is true for the set \(S(f)\), but with Lipschitz surfaces of codimension 2. In the case \(X=\mathbb R^ 2\) they obtain complete characterization of the sets of the form \(S_{+}(f)\) for a real Lipschitz function on \(X\) having one-sided directional derivatives for all \(x\in X\), and in the case \(X=\mathbb R^ 3\) for the sets of the form \(S(f)\), where \(f\) is a real Lipschitz function on \(\mathbb R^ 3\) having two-sided directional derivatives for all \(x\in \mathbb R^ 3\). They show by some examples constructed in Section 5 that the obtained results are, in some sense, the best possible.
scientific article; zbMATH DE number 2084039