On continuous but differentiable nowhere solutions in the moment problem

Topological genericity of nowhere differentiable functions in the disc algebra

In this paper we introduce a class of functions contained in the disc algebra $\mathcal{A}(D)$. We study functions $f \in \mathcal{A}(D)$, which have the property that the continuous periodic function $u = Ref|_{\mathbb{T}}$, where $\mathbb{T}$ is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function $f \in \mathcal{A}(D)$ has the above property. Afterwards, we strengthen this result by proving that, generically, for every function $f \in \mathcal{A}(D)$, both continuous periodic functions $u=Ref|_\mathbb{T}$ and $\tilde{u} =…

Nowhere differentiable functions of analytic type on products of finitely connected planar domains

Using Laurent decomposition and Mergelyan's theorem combined with Baire's category theorem, we prove generic nowhere differentiability on the distinguished boundary of functions of analytic type on products of planar domains bounded by finitely many disjoint Jordan curves. The parametrization of the boundaries are those induced by the natural Riemann maps.

连续但处处不可微的多元函数

Continuous nowhere differentiable multivariate functions

本文研究多元连续函数的强处处不可微性，通过复合单位速度曲线刻画，并证明在 Baire 范畴意义下，几乎所有连续函数都是强处处不可微的。

Some geometric properties of Riemann's non-differentiable function

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.

Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

In this paper we examine functions in the disc algebra $\mathcal{A}(D)$ and the polydisc algebra $\mathcal{A}(D^I)$, where $I$ is a finite or countably infinite set. We prove that, generically, for every $f \in \mathcal{A}(D)$ the continuous periodic functions $u=Ref|_{\mathbb{T}}$ and $\tilde{u} = Imf|_{\mathbb{T}}$ are nowhere differentiable on the unit circle $\mathbb{T}$. Afterwards, we generalize this result by proving that, generically, for every $f \in \mathcal{A}(D^I)$, where $I$ is as above, the continuous periodic functions $u=Ref|_{\mathbb{T}^I}$ and $\tilde{u} = Imf|_{\mathbb{T}^I}…

Nowhere differentiable functions with respect to the position

Let $Ω$ be a bounded domain in $\mathbb{C}$ such that $\partial Ω$ does not contain isolated points. Let $R(Ω)$ be the space of uniform limits on $\overlineΩ$ of rational functions with poles off $\overlineΩ$, endowed with the supremum norm. We prove that either generically all functions $f$ in $R(Ω)$ satisfy % $$ \limsup_{\substack{z \to z_0 z \in \partial Ω}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = + \infty $$ for every $z_0 \in \partial Ω$ or no such function in $R(Ω)$ meets this requirement. In the first case, the generic function $f \in R(Ω)$ is nowhere differentiable on $\partial Ω$ …

Non approximate derivability of the Takagi function

The Takagi function is a classical example of a continuous nowhere differentiable function. In this paper we prove that it is nowhere approximately derivable.

On one nearly everywhere continuous and nowhere differentiable function, that defined by automaton with finite memory

This paper is devoted to the investigation of the following function $$ f: x=Δ^{3}_{α_{1}α_{2}...α_{n}...}{\rightarrow} Δ^{3}_{φ(α_{1})φ(α_{2})...φ(α_{n})...}=f(x)=y, $$ where $φ(i)=\frac{-3i^{2}+7i}{2}$, $ i \in N^{0}_{2}=\{0,1,2\}$, and $Δ^{3}_{α_{1}α_{2}...α_{n}...}$ is the ternary representation of $x \in [0;1]$. That is values of this function are obtained from the ternary representation of the argument by the following change of digits: 0 by 0, 1 by 2, and 2 by 1. This function preserves the ternary digit $0$. Main mapping properties and differential, integral, fractal properties of the …

A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

We present a class of functions $\mathcal{K}$ in $C^0(\R)$ which is variant of the Knopp class of nowhere differentiable functions. We derive estimates which establish $\mathcal{K} \sub C^{0,\al}(\R)$ for $0<\al<1$ but no $K \in \mathcal{K}$ is pointwise anywhere improvable to $C^{0,\be}$ for any $\be>\al$. In particular, all $K$'s are nowhere differentiable with derivatives singular distributions. $\mathcal{K}$ furnishes explicit realizations of the functional analytic result of Berezhnoi. Recently, the author and simulteously others laid the foundations of Vector-Valued Calculus of Variation…

Level Sets of Differentiable Functions of Two Variables with Non-vanishing Gradient

We show that if the gradient of $f:\RR^2\rightarrow\RR$ exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set $\{x\in\RR^2:f(x)=c\}$ is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point. The second case holds only at the points of a discrete set. We also investigate the global structure of the level sets.

Superdifferential of the Takagi function

The Takagi function is a classical example of a continuous nowhere differentiable function. It has empty subdifferential except in a countable set where its subdifferential is $\mathbb{R}$. In this paper we characterize its superdifferential.

Differentiability and Hölder spectra of a class of self-affine functions

This paper studies a large class of continuous functions $f:[0,1]\to\mathbb{R}^d$ whose range is the attractor of an iterated function system $\{S_1,\dots,S_{m}\}$ consisting of similitudes. This class includes such classical examples as Pólya's space-filling curves, the Riesz-Nagy singular functions and Okamoto's functions. The differentiability of $f$ is completely classified in terms of the contraction ratios of the maps $S_1,\dots,S_{m}$. Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) $f$ is nowhere differentiable; (ii) $f$ is non-differentiable almost e…

On the properties of functions of the Takagi power class

By construction, functions of Takagi power class are similar to Takagi's continuous nowhere differentiable function. These functions have one real parameter $p>0$. They are defined by the series $S_p(x) = \sum_{n=0}^\infty (T_0(2^nx)/2^n)^p$, where $T_0(x)$ is the distance from the real number $x$ to the nearest integer. We study such properties of functions $S_p(x)$ as continuity, generalized Hölder condition, global maxima and differentiability at the points $x=1/3$ and $x=2/3$, for various values of $p$.

Intermittency of Riemann's non-differentiable function through the fourth-order flatness

Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, a…

Knot points of typical continuous functions

It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.