Solution to the gradient problem of C. E. Weil. (English) Zbl 1116.26007

The author gives a complete answer to the gradient problem of C. E. Weil by means of a counter-example. Precisely, he has found a differentiable function \(\;f:G\to R\;\), with \(\;G\subset R^2\) open, and an open set \(\Omega_1\subset R^2\;\) for which there exists a point \(p\in G\;\) such that \( \;\nabla f(p)\in \Omega_1\;\) but for a.e. \(\;q\in G\;\) the gradient \( \nabla f(q)\notin \Omega_1.\;\) This example also proves that the Denjoy-Clarkson property does not hold in higher dimensions.


26B05 Continuity and differentiation questions
28A75 Length, area, volume, other geometric measure theory
37E99 Low-dimensional dynamical systems
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