A new proof of Fréchet differentiability of Lipschitz functions
As well known, if \(f\) is a real valued Lipschitz function from an open set \(G\) (in a separable Banach space), then \(f\) is Gateaux differentiable in a subset \(D\) whose complement in \(G\) is a Gauss null set. Let \(A=\{f'(x)\); \(x\in D\}\). Assuming that the \(w^*\)-closure of \(A\) is norm separable, it is proved that \(G\) contains a point of Fréchet differentiability of \(f\). Moreover, for every segment \([u,v]\subset G\) and every \(m<f(v) -f(u)\), there is a point \(x\) of Fréchet differentiability of \(f\) such that \(f'(x)(v-u) >m\). The argument is self-contained and exhibits an algorithm for producing a sequence of points \((x_n)_n\) which converge to a point of Fréchet differentiability.
scientific article; zbMATH DE number 1529495