Non-Lipschitz functions with bounded gradient and related problems

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.