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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)

26B05 Continuity and differentiation questions