Continuous nowhere Hölder functions on ℤ_{𝕡}

Araújo, G.; Bernal-González, L.; Fernández-Sánchez, J.; Seoane–Sepúlveda, J.

doi:10.1090/proc/16185

Continuous nowhere Hölder functions on $\mathbb {Z}_{p}$
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by G. Araújo, L. Bernal-González, J. Fernández-Sánchez and J. B. Seoane–Sepúlveda

Proc. Amer. Math. Soc. 151 (2023), 1031-1040

DOI: https://doi.org/10.1090/proc/16185

Published electronically: December 21, 2022

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Abstract:

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on $[0,1]$ that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of $\mathbb {R}$, but in subsets of the $p$-adic field $\mathbb {Q}_p$.

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