A continuum dimensional algebra of nowhere differentiable functions

TL;DR

This paper constructs a large algebra of functions that are nowhere differentiable, revealing complex structures and relationships among such functions and their differentiability properties.

Contribution

It introduces a continuum-dimensional algebra of nowhere differentiable functions and explores the differentiability characteristics within generated algebras.

Findings

Constructed an algebra of dimension continuum of nowhere differentiable functions.

Identified that some generated algebras contain functions with limited differentiability points.

Showed that well-known nowhere differentiable functions can generate algebras with specific differentiability properties.

Abstract

We construct an algebra of dimension $2^{\aleph_0}$ consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.