Revista Matemática Complutense

, Volume 30, Issue 2, pp 269–279 | Cite as

Non-Lipschitz differentiable functions on slit domains

  • L. Bernal-González
  • P. Jiménez-Rodríguez
  • G. A. Muñoz-Fernández
  • J. B. Seoane-SepúlvedaEmail author


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.


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

Mathematics Subject Classification

Primary 26B35 Secondary 15A03 31C05 



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


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Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  • L. Bernal-González
    • 1
  • P. Jiménez-Rodríguez
    • 2
  • G. A. Muñoz-Fernández
    • 3
  • J. B. Seoane-Sepúlveda
    • 4
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de SevillaSevilleSpain
  2. 2.Department of Mathematical SciencesMathematics and Computer Science BuildingKentUSA
  3. 3.Departamento de Análisis Matemático, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  4. 4.ICMAT and Departamento de Análisis Matemático, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

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