JournalsrmiVol. 21, No. 3pp. 889–910

Solution to the gradient problem of C. E. Weil

  • Zoltán Buczolich

    Eötvös Loránd University, Budapest, Hungary
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Abstract

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set GR2G\subset \mathbb{R}^{2} we construct a differentiable function f:GRf:G\to\mathbb{R} for which there exists an open set Ω1R2\Omega_{1}\subset\mathbb{R}^{2} such that f(p)Ω1\nabla f(\mathbf{p})\in \Omega_{1} for a pG\mathbf{p}\in G but f(q)∉Ω1\nabla f(\mathbf{q})\not\in\Omega_{1} for almost every qG\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