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

[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.

[发布日期] 2010-12-15

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