Acta Mathematica Hungarica

, Volume 157, Issue 2, pp 371–386 | Cite as

Inverse images of open sets under gradient mapping

  • A. SkálováEmail author


For every pair of sets \({F, U \subset \mathbb{R}^d}\), \({d \geq 2}\), F being of Borel class Fσ and U being nonempty, bounded and open, we construct a Fréchet differentiable function \({f \colon \mathbb{R}^{d} \to \mathbb{R}}\) such that \({F \subset (\nabla{f})^{-1}(U)}\) and the Hausdorff dimension of \({(\nabla{f})^{-1}(U) \setminus F}\) does not exceed 1. Moreover \({(\nabla{f})(\mathbb{R}^d) \subset \overline{U}}\). This generalizes both Zelený [10] and Deville–Matheron [8] results about the properties of open sets preimages under the gradient mapping.

Key words and phrases

Denjoy–Clarkson property gradient Hausdorff measure reduction 

Mathematics Subject Classification

26B05 28A75 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

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