Further Notes on a Family of Continuous, Non-differentiable Functions

TL;DR

This paper investigates a family of continuous functions that are often nowhere differentiable, analyzing their derivatives and fractal dimensions to understand their complex behavior across different parameters.

Contribution

It provides new insights into the differentiability properties and fractal dimensions of a parameterized family of continuous functions.

Findings

Some functions are nowhere differentiable.

Hausdorff dimension of graphs varies with parameter a.

Differentiability behavior depends on the parameter value.

Abstract

We examine a parameterized family of functions F_a, all of which are continuous and some of which are nowhere or almost nowhere differentiable, we explore the behavior of F'_a and F"_a almost everywhere for different values of a, focusing on specific questions regarding F_a's differentiability for certain a, and we calculate the Hausdorff dimension of the graphs of all F_a.