# Solution to the gradient problem of C. E. Weil

### Zoltán Buczolich

Eötvös Loránd University, Budapest, Hungary

## 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 $\nabla f(\mathbf{q})\not\in\Omega_{1}$ for almost every $\mathbf{q}\in G$. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.

## Cite this article

Zoltán Buczolich, Solution to the gradient problem of C. E. Weil. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 889–910

DOI 10.4171/RMI/439