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Besicovitch Functions

  • Marek Jarnicki
  • Peter Pflug
Part of the Springer Monographs in Mathematics book series (SMM)

Summary

In Chap.  7, it was shown that \(\boldsymbol{\mathcal{B}}(\mathbb{I})\) is of first category in \(\mathcal{C}(\mathbb{I})\), i.e., most functions in \(\mathcal{C}(\mathbb{I})\) have somewhere on \(\mathbb{I}\) an infinite one-sided derivative. In the first part of this chapter, the construction of concrete functions belonging to \(\boldsymbol{\mathcal{B}}\boldsymbol{\mathcal{M}}(\mathbb{I})\) is discussed. The remaining part deals with a categorial argument proving that the set \(\boldsymbol{\mathcal{B}}\boldsymbol{\mathcal{M}}(\mathbb{I})\) is in some sense even a large set.

References

  1. [Bes24]
    A.S. Besicovitch, An investigation of continuous functions in connection with the question of their differentiability (Russian). Mat. Sb. 31, 529–556 (1924)Google Scholar
  2. [Mal84]
    J. Malý, Where the continuous functions without unilateral derivatives are typical. Trans. Am. Math. Soc. 283, 169–175 (1984)MathSciNetCrossRefGoogle Scholar
  3. [Mor38]
    A.P. Morse, A continuous function with no unilateral derivatives. Trans. Am. Math. Soc. 44, 496–507 (1938)MathSciNetCrossRefGoogle Scholar
  4. [Pep28]
    E.D. Pepper, On continuous functions without a derivative. Fundam. Math. 12, 244–253 (1928)CrossRefGoogle Scholar
  5. [Sin41]
    A.N. Singh, On functions without one-sided derivatives. Proc. Benares Math. Soc. 3, 55–69 (1941)MathSciNetGoogle Scholar
  6. [Sin43]
    A.N. Singh, On functions without one-sided derivatives ii. Proc. Benares Math. Soc. 4, 95–108 (1943)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Marek Jarnicki
    • 1
  • Peter Pflug
    • 2
  1. 1.Institute of MathematicsJagiellonian UniversityKrakówPoland
  2. 2.Insitute for MathematicsCarl von Ossietzky University OldenburgOldenburgGermany

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