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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.

##### MSC:
 26B05 Continuity and differentiation questions 28A75 Length, area, volume, other geometric measure theory 37E99 Low-dimensional dynamical systems
##### Keywords:
Denjoy-Clarkson property; Lebesgue measure
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##### References:
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