The set of gradients of a bump
Jan Kolár and Jan Kristensen
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Submission date: 02. Aug. 2002
MSC-Numbers: 26B05, 26B30, 46G05, 46B20
Keywords and phrases: gradient, range of derivative, bump, morse-sard theorem, critical set
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This is the first in a series of two papers.
The range of the gradient of a differentiable real-valued function with a non-empty and bounded support (i.e., a bump) is investigated. For a smooth bump on it is shown that the interior is connected and dense in . A purely topological characterization of such gradient ranges is however impossible. We give an example of a compact set that is homeomorphic to the closed unit disk, but such that no affine image of K is the gradient range of a smooth bump on . For smooth bumps on we show that the gradient range cannot be separated by a totally disconnected set. The proof relies on a Morse-Sard type result involving irreducible separators of . Proofs are carried out for a class of functions containing all those whose first order derivatives are Lipschitz or of bounded variation.
Finally, we present an example of a -smooth bump on , which has a gradient range with non-empty and disconnected interior, and a -smooth weak bump on with the same property.