Revista Matemática Iberoamericana

Solution to the gradient problem of C.E. Weil

Zoltán Buczolich

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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.

Article information

Rev. Mat. Iberoamericana Volume 21, Number 3 (2005), 889-910.

First available in Project Euclid: 11 January 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B05: Continuity and differentiation questions
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 37E99: None of the above, but in this section

gradient Denjoy-Clarkson property Lebesgue measure


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

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