Non-Lipschitz functions with bounded gradient and related problems

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• 作者：P. Jim&eacute ; nez-Rodrí ; guez¹ ; ^{pablo.jimenez.rod@gmail.com} ; G.A. Muñ ; oz-Ferná ; ndez² ; ^{gustavo_fernandez@mat.ucm.es} ; J.B. Seoane-Sepú ; lveda ; ² ; ^{jseoane@mat.ucm.es}
• 关键词：15A03 ; 26B05
• 刊名：Linear Algebra and its Applications
• 出版年：2012
• 出版时间：15 August, 2012
• 年：2012
• 卷：437
• 期：4
• 页码：1174-1181
• 全文大小：662 K

文摘

Let E be a topological vector space and let us consider a property . We say that the subset M of E formed by the vectors in E which satisfy is -lineable (for certain cardinal , finite or infinite) if contains an infinite dimensional linear space of dimension . In this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties: (1) Being continuous on , a.e. differentiable, with a.e. bounded derivative, and not Lipschitz. (2) Differentiable in 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.