The improper infinite derivatives of Takagi's nowhere-differentiable function

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:372

Article

Allaart, Pieter C.¹  Kawamura, Kiko¹ 

[1] Univ N Texas, Dept Math, Denton, TX 76203 USA

关键词: Takagi's function;    Nowhere-differentiable function;    Improper derivative;    Modulus of continuity;   

DOI  :  10.1016/j.jmaa.2010.06.059

来源: Elsevier

【 摘 要 】

Let T be Takagi's continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono [Acta Math. Hungar. 49 (1987) 315-324], we give a complete characterization of those points where T has a left-sided, right-sided, or two-sided infinite derivative. This characterization is illustrated by several examples. A consequence of the main result is that the sets of points where T'(x) = +/-infinity have Hausdorff dimension one. As a byproduct of the method of proof, some exact results concerning the modulus of continuity of T are also obtained. (C) 2010 Elsevier Inc. All rights reserved.

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