## Real Analysis Exchange

### A note on the gradient problem

#### Abstract

C. E. Weil 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 $^N$ to $\mathbb{R}^N$. Let $G \subset \mathbb{R}^N$ be a nonempty open set and let $g^{-1}(G) \ne \emptyset.$ Does $g^{-1}(G)$ have positive $N$-dimensional Lebesgue measure?” For $N = 1$ the answer is yes as was first proved by Denjoy in 1916. Z. Buczolich gave a partial answer to this problem showing that $g^{-1}(G)$ has positive one-dimensional Hausdorff measure. In other words, he proved that the gradient has the “one-dimensional Denjoy-Clarkson property”. In the present article, we prove the Buczolich result using a quite different method. Moreover, our method gives improvements and generalizations.

#### Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 225-235.

Dates
First available in Project Euclid: 1 June 2012

https://projecteuclid.org/euclid.rae/1338515217

Mathematical Reviews number (MathSciNet)
MR1433610

Zentralblatt MATH identifier
0879.26041

Subjects
Primary: 26B05: Continuity and differentiation questions

Keywords

#### Citation

Holický, P.; Malý, J.; Zajíček, L.; Weil, C. E. A note on the gradient problem. Real Anal. Exchange 22 (1996), no. 1, 225--235. https://projecteuclid.org/euclid.rae/1338515217

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