The higher order partial derivatives of Okamoto's function with respect to the parameter

TL;DR

This paper studies the higher order partial derivatives of Okamoto's family of self-affine functions, analyzing their fractal dimensions, differentiability properties, and points of infinite derivative, revealing new phenomena beyond known properties.

Contribution

It introduces and analyzes the higher order derivatives of Okamoto's functions, providing new insights into their fractal and differentiability characteristics.

Findings

Computed the box-counting dimension of the derivatives' graphs

Characterized the differentiability of the derivatives

Identified points where derivatives are infinite

Abstract

Let $\{F_a: a\in(0,1)\}$ be Okamoto's family of continuous self-affine functions, introduced in [{\em Proc. Japan Acad. Ser. A Math. Sci.} {\bf 81} (2005), no. 3, 47--50]. This family includes well-known ``pathological" examples such as Cantor's devil's staircase and Perkins' continuous but nowhere differentiable function. It is well known that $F_a(x)$ is real analytic in $a$ for every $x\in[0,1]$. We introduce the functions \[ M_{k,a}(x):=\frac{\partial^k}{\partial a^k}F_a(x), \qquad k\in\mathbb{N}, \quad x\in[0,1]. \] We compute the box-counting dimension of the graph of $M_{k,a}$, characterize its differentiability, and investigate in detail the set of points where $M_{k,a}$ has an infinite derivative. While some of our results are similar to the known facts about Okamoto's function, there are also some notable differences and surprising new phenomena that arise when considering the higher order partial derivatives of $F_a$.