Solution to the gradient problem of C. E. Weil

Abstract

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set $G\subset {\mathbb{R}}^2$ we construct a differentiable function $f:G\to \mathbb{R}$ for which there exists an open set ${\Omega }_1\subset {\mathbb{R}}^2$ such that $\nabla f(\mathbf{p})\in {\Omega }_1$ for a $\mathbf{p}\in G$ but $\nabla f(\mathbf{q})\notin {\Omega }_1$ for almost every $\mathbf{q}\in G$. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.