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A note on the gradient problem. (English) Zbl 0879.26041
C. E. Weil [Real Anal. Exch. 16, 373 (1991)] formulated the following problem: “Assume that $$f$$ is a differentiable real-valued function of $$N$$ real variables, $$N\geq 2$$, and let $$g=\nabla f$$ denote its gradient, which is a function from $$R^N$$ to $$R^N$$. Let $$G\subset R^N$$ be a nonempty open set and let $$g^{-1}(G)\neq \emptyset$$. Does $$g^{-1}(G)$$ have positive $$N$$-dimensional Lebesgue measure?” For $$N=1$$ the answer is yes as was first proved by A. Denjoy [Enseign. Math. 18, 320-328 (1916; JFM 46.0381.05)]. Z. Buczolich [Real Anal. Exch. 18, No. 1, 221-224 (1993; Zbl 0783.26010)] gave a partial answer to this problem showing that $$g^{-1}(G)$$ has positive one-dimensional Hausdorff measure. This result is generalized in this paper. It is proved that any one-dimensional projection of $$g^{-1}(G)$$ is of positive one-dimensional Hausdorff measure. The authors also prove that $$g^{-1}(G)$$ is not a $$\sigma$$-porous set and that $$g^{-1}(G)$$ is porous at none of its points.
Reviewer: D.Medková (Praha)

MSC:
 26B05 Continuity and differentiation questions