Non-Lipschitz differentiable functions on slit domains
It is proved the existence of large algebraic structures—including large vector subspaces or infinitely generated free algebras—inside the family of non-Lipschitz differentiable real functions with bounded gradient defined on special non-convex plane domains. In particular, this yields that there are many differentiable functions on plane domains that do not satisfy the mean value theorem.
KeywordsNon-Lipschitz function Differentiable function Domain in the plane Free algebra
Mathematics Subject ClassificationPrimary 26B35 Secondary 15A03 31C05
We are indebted to our colleague José Antonio Prado-Bassas for drawing Fig. 1.
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