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Non-Lipschitz functions with bounded gradient and related problems

Jiménez Rodríguez, Pablo and Muñoz Fernández, Gustavo Adolfo and Seoane Sepúlveda, Juan Benigno (2012) Non-Lipschitz functions with bounded gradient and related problems. Linear Algebra and its Applications, 437 (4). pp. 1174-1181. ISSN 0024-3795

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Abstract

Let E be a topological vector space and let us consider a property P. We say that the subset M of E formed by the vectors in E which satisfy P is μ-lineable (for certain cardinal μ, finite or infinite) if M ∪ {0} contains an infinite dimensional linear space of dimension μ. In this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties:(1) Being continuous on [0, 1], a.e. differentiable, with a.e. bounded derivative, and not Lipschitz. (2) Differentiable in (R2)R and not enjoying the Mean Value Theorem. (3) Real valued differentiable on an open, connected, and non-convex set, having bounded gradient,non-Lipschitz, and (therefore) not verifying the Mean Value Theorem.


Item Type:Article
Uncontrolled Keywords:Lineability; Spaceability; Continuous non-Lipschitzfunctions; Mean Value Theorem
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:16318
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Deposited On:11 Sep 2012 09:38
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