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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
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