In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in P is nowhere differentiable. The class P naturally appear from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to P if and only if some geometrical inequality holds for a family of parabolas with vertexes on this function. As its application, we study the behavior of the Hamilton Jacobi flow starting from a function in P. A connection between P and some functional series is also investigated. In terms of second-order central differences, we give a necessary and sufficient condition so that a function given by the series belongs to P. This enables us to construct a large number of examples of functions in P through an explicit formula.
In this paper, we prove that for some Generalized Takagi Classes, in particular for the Takagi-Van der Waerden Class, the functions are nowhere differentiable if, and only if, the sequence of weights does not belong to $c_0$.
Let T be Takagi's continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono, we give a complete characterization of those points where T has a left-sided, right-sided, or two-sided infinite derivative. This characterization is illustrated by several examples. A consequence of the main result is that the sets of points where T'(x) is infinite have Hausdorff dimension one. As a byproduct of the method of proof, some exact results concerning the modulus of continuity of T are also obtained.
We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.
We will give an elementary nonstandard proof that the family of generalized blancmange functions are nowhere differentiable. The proof follows from the intuitive characterization of differentiability at a point as almost $δ$ affine along with the transfer of the functional equations these functions satisfy. We also give elementary nonstandard proofs of the uniform density of these functions among continuous functions. Finally, we discuss work done with the Python programming language in displaying these functions.
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} =…
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.
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.
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}…
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 $\partial Ω$ …
The Takagi function is a classical example of a continuous nowhere differentiable function. In this paper we prove that it is nowhere approximately derivable.
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 …
Using a few basics from integration theory, a short proof of nowhere-differentiability of Weierstrass functions is given. Restated in terms of the Fourier transformation, the method consists in principle of a second microlocalisation, which is used to derive two general results on existence of nowhere differentiable functions. Examples are given in which the frequencies are of polynomial growth and of almost quadratic growth as a borderline case.
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…
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.
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.
A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It is shown, e.g., that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that Hausdorff dimension of the graph of such a function is 1.
We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the incremental quotients are bounded has maximal Hausdorff dimension.
In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of fractional order alpha of function f(x) is defined as a fractional integration of order (n-alpha) of the derivative f^(n)(x) of integer order n. The requirement of the existence of integer-order derivatives allows us to conclude that the local fractional derivative cannot be considered as the best method to describe nowhere diff…
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…
Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this docume…
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 $p$.
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…
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.
We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense.
The functions of the Takagi exponential class are similar in construction to the continuous, nowhere differentiable Takagi function described in 1901. They have one real parameter $v\in (-1;1)$ and at points $x\in{\mathbb R}$ are defined by the series $T_v(x) = \sum_{n=0}^\infty v^n T_0(2^nx)$, where $T_0(x)$ is the distance between $x$ and the nearest integer point. If $v=1/2$ then $T_v$ coincides with Takagi's function. In this paper, for different values of the parameter $v$, we study the global extremes of the functions $T_v$, as well as the sets of extreme points. All functions of $T_v$ h…
We construct rich vector spaces of continuous functions with prescribed curved or linear pathwise quadratic variations. We also construct a class of functions whose quadratic variation may depend in a local and nonlinear way on the function value. These functions can then be used as integrators in Föllmer's pathwise Itō calculus. Our construction of the latter class of functions relies on an extension of the Doss--Sussman method to a class of nonlinear Itō differential equations for the Föllmer integral. As an application, we provide a deterministic variant of the support theorem for diffusion…
We introduce a class $Λ_{s}$ of functions with complicated local structure. Any function from the class belongs to one of three specifically defined types $f^s _k$, $f_+$, and $f^{-1} _+$ or is a specifically defined composition of two or three functions of these types. Differential, integral, fractal and other properties of such functions are investigated. In particular, all functions $f$ from $Λ_{s}$ such that $f(x)\ne x$, $f(x)\ne -\frac{s-1}{s+1}-x$ and $f(x)\ne 1-x$ are nonmonotonic and nowhere differentiable. The Hausdorff--Besicovitch dimension of a plot of each $f \in Λ_s$ is equal to …
