## 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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