The set of continuous nowhere differentiable functions

The Set of Continuous Nowhere Differentiable Functions

Let C be the space of all real-valued continuous functions defined on the unit interval provided with the uniform norm. In the Scottish Book, Banach raised the question of the descriptive class of the subset D of C consisting of all functions which are differentiable at each point of [0,1]. Banach pointed out that D forms a coanalytic subset of C and asked whether D is a Borel set. Later Mazurk...

Everywhere Continuous Nowhere Differentiable Functions

Here I discuss the use of everywhere continuous nowhere differentiable functions, as well as the proof of an example of such a function. First, I will explain why the existence of such functions is not intuitive, thus providing significance to the construction and explanation of these functions. Then, I will provide a specific detailed example along with the proof for why it meets the requireme...

The Prevalence of Continuous Nowhere Differentiable Functions

In the space of continuous functions of a real variable, the set of nowhere dilferentiable functions has long been known to be topologically "generic". In this paper it is shown further that in a measure theoretic sense (which is different from Wiener measure), "almost every" continuous function is nowhere dilferentiable. Similar results concerning other types of regularity, such as Holder cont...

An Analysis of Katsuura’s Continuous Nowhere Differentiable Function

Nowhere Differentiable Curves

We consider a family of smooth maps on an infinite cylinder which have invariant curves that are nowhere smooth. Most points on such a curve are buried deep within its spiked structure, and the outermost exposed points of the curve constitute an invariant subset which we call the "facade" of the curve. We find that for surprisingly many of the maps in the family, all points in the facades of th...