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

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)], 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) = ±∞ 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. AMS 2000 subject classification: 26A27 (primary); 26A15 (secondary) ∗Supported in part by Japanese GCOE Program G08: “Fostering Top Leaders in Mathematics Broadening the Core and Exploring New Ground”. †Address: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, USA; E-mail: [email protected], [email protected]