Porosity and the Darboux property of Fréchet derivatives
Two theorems are proved which are converses in the case of Euclidean spaces of existing theorems in Banach spaces. The new theorems are: Theorem 1.3. Let \(B\subset \mathbb R^d\) be nonempty without isolated points such that the interior of \(B\) is connected. Then the following assertions are equivalent: (i) \(\mathbb R^d\backslash B\) is porous at every \(a\in B\cup\partial B\). (ii) The graph of \(g\) is connected whenever \(g\) is a Fréchet derivative of a function \(f: B\to \mathbb R\) on \(B\). (iii) \(g(B)\) is connected whenever \(g\) is a Fréchet derivative of a function \(f: B\to \mathbb R\) on \(B\). Proposition 4.2. Let \(a\in \mathbb R^d\) and \(E\subset \mathbb R^d\backslash\{a\}\) be a set which is not porous at \(a\). Then there is a Lipschitz function \(f: \mathbb R^d\to \mathbb R\), Fréchet differentiable on \(\mathbb R^d\backslash E\), such that \(f^{\prime}(a)=0\) and \(|f^{\prime}(u|\geq1\) for any \(u\in \mathbb R^d\backslash(E\cup \{a\})\).
scientific article; zbMATH DE number 5652539