On maxima of Takagi-van der Waerden functions

Abstract:

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