The partial derivative of Okamoto's functions with respect to the parameter

TL;DR

This paper investigates the partial derivative of Okamoto's functions with respect to their parameter, focusing on the case a=1/3, revealing properties of a nowhere differentiable function with a measure zero set of infinite derivatives.

Contribution

It provides a detailed analysis of the parameter derivative of Okamoto's functions, especially at a=1/3, highlighting the fractal nature of the set of points with infinite derivatives.

Findings

The set of points with infinite derivative has Hausdorff dimension 1.

The function exhibits nowhere differentiability at a=1/3.

The measure of the set of infinite derivatives is zero.

Abstract

The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.