Non-Lipschitz differentiable functions on slit domains

Abstract

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.

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Fig. 1

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Acknowledgements

We are indebted to our colleague José Antonio Prado-Bassas for drawing Fig. 1.

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Correspondence to J. B. Seoane-Sepúlveda.

Additional information

L. Bernal-González: supported by the Plan Andaluz de Investigacin de la Junta de Andaluca FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P.

P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlvedal: supported by the Spanish Ministry of Science and Innovation, Grant MTM2015-65825-P.

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Bernal-González, L., Jiménez-Rodríguez, P., Muñoz-Fernández, G.A. et al. Non-Lipschitz differentiable functions on slit domains. Rev Mat Complut 30, 269–279 (2017). https://doi.org/10.1007/s13163-016-0218-x

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Keywords

  • Non-Lipschitz function
  • Differentiable function
  • Domain in the plane
  • Free algebra

Mathematics Subject Classification

  • Primary 26B35
  • Secondary 15A03
  • 31C05