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Jiménez Rodríguez, Pablo and Muñoz-Fernández, Gustavo A. and Seoane-Sepúlveda, Juan B.
(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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Official URL: http://www.sciencedirect.com/science/article/pii/S0024379512002741#

## 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 |
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Uncontrolled Keywords: | Lineability; Spaceability; Continuous non-Lipschitzfunctions; Mean Value Theorem |

Subjects: | Sciences > Mathematics > Mathematical analysis |

ID Code: | 16318 |

Deposited On: | 11 Sep 2012 09:38 |

Last Modified: | 29 Nov 2016 09:08 |

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