%0 Journal Article
%A Jiménez Rodríguez, Pablo
%A Muñoz-Fernández, Gustavo A.
%A Seoane-Sepúlveda, Juan B.
%T Non-Lipschitz functions with bounded gradient and related problems
%D 2012
%@ 0024-3795
%U https://hdl.handle.net/20.500.14352/42298
%X 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.
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