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Baire Category Approach

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Book cover Continuous Nowhere Differentiable Functions

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Summary

While in Part I, some concrete functions were discussed, this chapter shows how Baire category methods lead to a description of typical continuous functions on the interval \(\mathbb{I} = [0,1]\). In Sect. 7.2, we prove that most (in the categorial sense) of the continuous functions on \(\mathbb{I}\) belong to \(\boldsymbol{\mathcal{N}}\boldsymbol{\mathcal{D}}_{\pm }(\mathbb{I})\), while in Sect. 7.5, it is shown that the set \(\boldsymbol{\mathcal{N}}\boldsymbol{\mathcal{D}}_{\pm }^{\infty }(\mathbb{I})\) of all continuous functions on \(\mathbb{I}\) having nowhere a unilateral (finite or infinite) derivative is a thin set (in the categorial sense). Nevertheless, later, in Sect. 11.1, we will see that \(\boldsymbol{\mathcal{N}}\boldsymbol{\mathcal{D}}_{\pm }^{\infty }(\mathbb{I})\) is not empty.

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Authors and Affiliations

  1. Institute of Mathematics, Jagiellonian University, Kraków, Poland

    Marek Jarnicki

  2. Insitute for Mathematics, Carl von Ossietzky University Oldenburg, Oldenburg, Germany

    Peter Pflug

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Jarnicki, M., Pflug, P. (2015). Baire Category Approach. In: Continuous Nowhere Differentiable Functions. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-12670-8_7

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