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- Title
Properties of the C <sup>1</sup>-smooth functions with nowhere dense gradient range.
- Authors
Korobkov, M. V.
- Abstract
One of the main results of the present article is as follows Theorem. Let v: Ω → ℝ be a C1-smooth function on a domain Ω ⊂ ℝ2. Suppose that Int∇v(Ω) = ∅. Then, for every point z ∈ Ω, there is a straight line L ∋ z such that ∇v ≡ const on the connected component of the set L ⋂ Ω containing z. Also, we prove that, under the conditions of the theorem, the range of the gradient ∇v(Ω) is locally a curve and this curve has tangents in the weak sense and the direction of these tangents is a function of bounded variation.
- Subjects
SMOOTHNESS of functions; MATHEMATICAL functions; ALGEBRA; MATHEMATICAL analysis; MATHEMATICAL research
- Publication
Siberian Mathematical Journal, 2007, Vol 48, Issue 6, p1019
- ISSN
0037-4466
- Publication type
Article
- DOI
10.1007/s11202-007-0104-3