The improper infinite derivatives of Takagiʹs nowhere-differentiable function

Let T be Takagiʹs 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 ) = ± ∞ 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.