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The $$n$$-dimensional gradient has the 1-dimensional Denjoy-Clarkson property. (English) Zbl 0783.26010
The author proves the following result:
Let $$f: \Omega\to\mathbb{R}$$ be a differentiable function, where $$\Omega\subset\mathbb{R}^ n$$ is open, then the set $$(\nabla f)^{-1}(G)$$ is either empty or has positive 1-dimensional Hausdorff measure.
The present result is connected to a conjecture of C. E. Weil [Real Anal. Exch. 16, No. 1, 373 (1991)].

##### MSC:
 26B35 Special properties of functions of several variables, Hölder conditions, etc. 28A78 Hausdorff and packing measures