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Gradient ranges of bumps on the plane

Authors:	Jan Kolá r Jan Kristensen

Institution:	Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic ; Mathematical Institute, 24-29 St Giles', University of Oxford, Oxford OX1 3LB, United Kingdom

Abstract:	For a $\mathcal{C}^1$ -smooth bump function $b \colon {\mathbb R}^{2} \to \mathbb{R}$ we show that the gradient range $\nabla b( {\mathbb R}^{2} )$ is the closure of its interior, provided that $\nabla b$ admits a modulus of continuity $\omega = \omega (t)$ satisfying $\omega (t)/\sqrt{t} \to 0$ as $t \searrow 0$ . The result is a consequence of a more general result about gradient ranges of bump functions $b \colon {\mathbb R}^{n} \to \mathbb{R}$ of the same degree of smoothness. For such bump functions we show that for open sets $G \subset {\mathbb R}^{n}$ , either the intersection $\nabla b( {\mathbb R}^{n}) \cap G$ is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.

Keywords:	Gradient range derivative bump Morse-Sard theorem

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