On maxima of Takagi-van der Waerden functions

Abstract

Generalizing Takagi’s function F 2 ( x ) {F_2}\left ( x \right ) and van der Waerden’s function F 10 ( x ) {F_{10}}\left ( x \right ) , we introduce a class of nowhere differentiable continuous functions F r ( x ) {F_r}\left ( x \right ) , r ⩾ 2 r \geqslant 2 . Some properties of F r ( x ) {F_r}\left ( x \right ) concerning especially maxima are discussed. When r r is even, the Hausdorff dimension of the set of x , {x^,} ’s giving the maxima of F r ( x ) {F_r}\left ( x \right ) is proved to be 1 / 2 1/2 .