Functions of two variables with large tangent plane sets
type: Item
instance of: scholarly article
title: Functions of two variables with large tangent plane sets
zbMATH Open document ID: 0915.26005
author: Zoltán Buczolich
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-07-07T00:00:00Z
review text: Main result: There exist a \(C^1\) function \(f: \mathbb{R}^2\to \mathbb{R}\), a nowhere dense closed set \(E\subseteq [0,1]\times [0,1]\) of zero two-dimensional measure, and a nonempty open set \(H\subseteq \mathbb{R}^3\) such that for any \((a,b,c)\in H\) we can find an \((x_0,y_0)\in E\) for which the equation of the tangent plane to the surface \(z= f(x,y)\) at \((x_0, y_0)\) is \(z= ax+ by+ c\). As a final remark, the author observes that the Hausdorff dimension of the set \(E\) is less than 2, but not less than 1. Open problem: Can the Hausdorff dimension of \(E\) be equal to 1? It is also proved that the theorem above (whose proof is very laborious) is no longer valid when \(C^1\) is replaced by \(C^2\). The relationship with the Denjoy-Young-Saks theorem (concerning the behavior almost everywhere of the derived numbers of an arbitrary real function of a real variable) is also investigated.
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 1218580
zbMATH Keywords: two-dimensional measure
zbMATH Keywords: \(C^1\) function
zbMATH Keywords: tangent plane
zbMATH Keywords: Hausdorff dimension
reviewed by: Solomon Marcus
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5848
OpenAlex ID: W1968153545
cites work: A property of derivatives
cites work: Differential Topology
cites work: A Dense Arcwise Connected Set of Critical Points - Molehills Out of Mountains
cites work: The behavior of a function on its critical set
cites work: The measure of the critical values of differentiable maps
DOI: 10.1006/JMAA.1997.5848
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description: scientific article; zbMATH DE number 1218580