The prevalence of continuous nowhere differentiable functions

Abstract

Abstract. 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 continuity, are discussed.

On differentiability properties of typical continuous functions and Haar null sets

“…Let D (D * ) be the set of all continuous functions f on [0, 1] which have a derivative f (x) ∈ R (f (x) ∈ R * , respectively) at least at one point x ∈ (0, 1). B. R. Hunt (1994) In the present article it is proved that neither D * nor its complement is Haar null in C[0, 1]. Moreover, the same assertion holds if we consider the approximate derivative (or the "strong" preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.…”

The Contribution of Wavelets in Multifractal Analysis

“…Since this is true for any γ > 0 we recover that the Hölder exponent of almost every continuous function vanishes everywhere, see [16].…”

“…Such results, where one proves that a property holds "almost surely" in a given function space E require the proof of results of this type, which hold for the sum of an arbitrary function f ∈ E and of a stochastic process whose sample path almost surely belong to E, see [12,16,17]. Here, the corresponding prevalent result is that the Hölder exponent of almost every continuous function is everywhere less than 1/2.…”

Properties of generic and almost every mappings in various nonlogically compact Polish abelian groups.

“…Thus the function gm is differentiable at almost every x E J; it follows that 9 is differentiable at almost every x E J. Then 9 is an element of the set of all 1 E C(~) such that 1 has finite derivative at at least one point, a set which is Haar null by Hunt's result in [25]. Since the complement of MT N I is contained in a…”

“…For example, consider the following pair of theorems concerning e([O, 1], lR), the space of continuous real-valued functions on the closed unit interval. The first is a classical result of Banach's, and the second more recent result is due to Hunt [25] Of course it is not always the case that a subset of a group is both comeager and co-Haar null. As another example, consider the space consisting of all permutations on N, denoted by SX!'…”