Typical Continuous Function Research Articles

On sets where $\operatorname{lip} f$ is finite

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called little lip function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r} \frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{\sigma \delta}$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost everywhere.

Graphs of continuous functions and packing dimension

We prove that for a typical continuous function f∈C(X) over an uncountable compact metric space X, the packing dimension of its graphGf={(x,f(x))|x∈X} is dimP⁡(X)+1; we also consider decompositions of functions in C([0,1]) in terms of upper box dimension as well as packing dimension, which are quite different from the case of Hausdorff dimension.

No-where averagely differentiable functions: Baire category and the Takagi function

A classical and well-known result due to Banach and Mazurkiewicz says that a typical (in the sense of Baire) continuous function on the unit interval is no-where differentiable. In this paper we prove that a typical (in the sense of Baire) continuous function \(f\) on [0, 1] is spectacularly more irregular than suggested by Banach and Mazurkiewicz’s result. Namely, not only is the difference quotient \(\frac{f(x+h)-f(x)}{h}\) divergent as \(h\rightarrow 0\) for all \(x\), but the function \(h\rightarrow \frac{f(x+h)-f(x)}{h}\) diverges so badly as \(h\rightarrow 0\), that it remains divergent even after being “smoothened out” using iterated Cesaro averages of arbitrary high order. More precisely, we introduce the notion of higher order average differentiability based on iterated Cesaro averages and prove that not only is a typical (in the sense of Baire) continuous function on [0, 1] no-where differentiable, but it is even no-where averagely differentiable of any order. We also show that the no-where differentiable Takagi function is, in fact, no-where averagely differentiable.

Coding discretizations of continuous functions

We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical continuous function when the diameter of the discretization tends to zero. Our main result is that any finite word appears on a subsequence discretization with any desired limit frequency.

Irregular 1-sets on the graphs of continuous functions

One can define in a natural way irregular 1-sets on the graphs of several fractal functions, like Takagi’s function, Weierstrass-Cellerier type functions and the typical continuous function. These irregular 1-sets can be useful during the investigation of level-sets and occupation measures of these functions. For example, we see that for Takagi’s function and for certain Weierstrass-Cellerier functions the occupation measure is singular with respect to the Lebesgue measure and for almost every level the level set is finite.

Limit sets of typical continuous functions

Given a metrizable compact topological n-manifold X with boundary and a finite positive Borel measure μ on X, we prove that for the typical continuous function f : X → X , it is true that for every point x in a full μ-measure subset of X the limit set ω ( f , x ) is a Cantor set of Hausdorff dimension zero, f maps ω ( f , x ) homeomorphically onto itself, each point of ω ( f , x ) has a dense orbit in ω ( f , x ) and f is non-sensitive at each point of ω ( f , x ) ; moreover, the function x → ω ( f , x ) is continuous μ-almost everywhere.

Hausdorff Measures of Different Dimensions are Isomorphic under the Continuum Hypothesis

We show that the Continuum Hypothesis implies that for every $ 0 < d_1 \leq d_2 < n $ the measure spaces $(\mathbb{R}^n,\mathcal{M}_{\mathcal{H}^{d_1}},\mathcal {H}^{d_1})$ and $(\mathbb{R}^n,\mathcal{M}_{\mathcal{H}^{d_2}},\mathcal{H}^{d_2})$ are isomorphic, where $\mathcal{H}^d$ is $d$-dimensional Hausdorff measure and $\mathcal{M}_{d}$ is the $\sigma$-algebra of measurable sets with respect to $\mathcal{H}^d$. This is motivated by the well-known question (circulated by D. Preiss) whether such an isomorphism exists if we replace measurable sets by Borel sets. We also investigate the related question whether every continuous function (or the typical continuous function) is Holder continuous (or is of bounded variation) on a set of positive Hausdorff dimension.

Chaos and the Recurrent Set

Let $f$ be an element of $C(I,I)$ with $R(f)=\{x\in I:x\in \omega (x,f)\}$ its recurrent set. We study the relationship between the structure of $R(f)$ and the chaotic nature of the function $f$ . We show that $R(f)$ is always a $G_{\delta }$ set whenever $f$ has zero topological entropy, although $R(f)$ is closed for the typical continuous function $f$ with zero topological entropy. We also develop necessary and sufficient conditions on $f$ for $R(f)$ to be closed.

Micro tangent sets of continuous functions

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set $A$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s function is graph like, and Weierstrass’s nowhere differentiable function is central graph like.

A Type of Path Derivative

In this paper a type of path derivative, which is not based on the non-empty path intersection, is introduced. Such derivatives share some basic properties with sequencial derivatives but there exist some sharp contrasts between them. The path derivative here restricts the speed of convergence in the definition of limits naturally and has a sharp contrast with the classical Dini derivatives for the typical continuous function.

ON DINI AND APPROXIMATE DINI DERIVATES OF TYPICAL CONTINUOUS FUNCTIONS

In the thirties, Banach, Mazurkiewicz and Jarn\'{\i}k found relations connecting Dini derivates of a typical continuous function on $[0,1]$ at all points of $(0,1)$. We prove, answering a question of K. M. Garg, that there are no further relations of this sort. An analogous result is proved also for approximate Dini derivates. The aim of this note is to present relatively simple proofs of these results. An article containing an improvement of these results in several directions (with a considerably more complicated proof) is in preparation.

On preponderant differentiability of typical continuous functions

In the literature, several definitions of a preponderant derivative exist. An old result of Jarník implies that a typical continuous function on $[0,1]$ has a (strong) preponderant derivative at no point. We show that a typical continuous function on $[0,1]$ has an infinite (weak) preponderant derivative at each point from a $c$-dense subset of $(0,1)$.

Dynamics of typical continuous functions

S. J. Agronsky, A. M. Bruckner, and M. Laczkovic have studied the behaviour of the sequence ( f n ( x ) ) ({f^n}(x)) where f is the typical continuous function from the closed unit interval I into itself and x the typical point of I. In particular, they have proved that the typical limit set ω ( f , x ) \omega (f,x) is a Cantor set of Menger-Uryson dimension zero. Using mainly the Tietze extension theorem, we have found a shorter proof of this result which applies to a more general situation. As a matter of fact, we have replaced the closed unit interval by a compact N-dimensional manifold and the Menger-Uryson dimension by the Hausdorff one. We have also proved that, for the typical continuous function f, the function x → ω ( f , x ) x \to \omega (f,x) is continuous at the typical point x. It follows that the typical limit set is not a fractal and that, for the typical continuous function f, the sequence ( f n ( x ) ) ({f^n}(x)) is not chaotic.

Структурные свойств а типичных непрерывн ых функций, определенных на метр ических локально ком пактных пространствах

LetC (E) be the space of real-valued continuous functions on a locally compact metric spaceE without isolated points, endowed with the uniform metric. The structure of sections of typical functions from C(E) in the sense of Baire's category is studied by functions of Lipschitz type. In particular, it is proved forE, which is generally not endowed by a linear structure, that a theorem is valid which, in the particular case ofE=[0, 1], implies a well-known result, due to Jarnik, on subdifferential numbers of a typical continuous function on a closed interval.

Density Continuity versus Continuity

Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if it is a nowhere dense Fσ set. The corresponding characterization for the approximately continuous functions is a AErst category Fσ set. An alternative proof of that result is given. Density continuous functions belong to the class Baire*1, unlike the approximately continuous functions.

On the periodic points of a typical continuous function

Let n and k be arbitrary natural numbers. We prove that for a continuous f, every neighborhood of any periodic point of f with n contains periodic points of f with n * k . The purpose of this paper is to examine the structure of the set of periodic points of continuous functions. We shall denote by C the set of continuous functions mapping the interval [0, 1] into itself. This set becomes a metric space with the supremum metric. By the term typical continuous function we mean that the set of all functions having the property under consideration is a residual subset of the metric space C. In [1] S. J. Agronsky, A. M. Bruckner, and M. Laczkovich proved that for a continuous f any neighborhood of a periodic point contains periodic points of large periods. In this paper we prove that this result is true if arbitrarily large periods is replaced by period k * n (k = 1 , 2, ...) (Theorem 1). As a consequence, for a continuous f we have that Pf the set of periodic points with n, is uncountable, dense in itself, but not closed. Also, P) is a residual subset of Fix(fn). In [1] it is proved that, for a continuous f, Fix(f n) is nowhere dense and perfect. In this paper we prove (Theorem 2) that is bilaterally strongly (D-porous. For f E C and n E N we define fn (x) by f '(x) = f(x) and fn(x) = f(fn-l(X)). PJn denotes the set of periodic points of f with n. We denote = {x: fn(X) = x}; then P = Fix(fn) Uk 0 denote B(f ,e) = {g E C: Ilf gll < }. Theorem 1. Let n , k E N be arbitrary. Then for a continuous f, every neighborhood of any periodic point of f with n contains periodic points of f with n * k. Received by the editors August 3, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20; Secondary 26A18. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

On the Periodic Points of a Typical Continuous Function

Let $n$ and $k$ be arbitrary natural numbers. We prove that for a typical continuous function $f$, every neighborhood of any periodic point of $f$ with period $n$ contains periodic points of $f$ with period $n \cdot k$.

Continuous functions with everywhere infinite variation with respect to sequences

We prove that if { a n } n = 1 ∞ \{ {a_n}\} _{n = 1}^\infty is such that a n ↘ 0 {a_n} \searrow 0 and \[ lim n → ∞ ( a n + 1 / a n ) = 1 , \lim \limits _{n \to \infty } ({a_n}_{ + 1}/{a_n}) = 1, \] then for the typical continuous function f f we have \[ S n 0 := ∑ n = n 0 ∞ | f ( x n + 1 ) − f ( x n ) | = + ∞ {S_{{n_0}}}: = \sum \limits _{n = {n_0}}^\infty | f({x_n}_{ + 1}) - f({x_n})| = + \infty \] whenever x ∈ [ 0 , 1 − a n 0 ] x \in [0,1 - {a_{{n_0}}}] and x n ∈ [ x + a n + 1 , x + a n ] {x_n} \in [x + {a_{n + 1}},x + {a_n}] . Based on our result in a previous paper, we know that the above theorem fails to hold if a n + 1 / a n = λ &gt; 1 {a_{n + 1}}/{a_n} = \lambda &gt; 1 . We also prove that if { a n } n = 1 ∞ \{ {a_n}\} _{n = 1}^\infty is such that a n ↘ 0 {a_n} \searrow 0 , then for the typical continuous function f f we have S n 0 = + ∞ if x n = x + a n {S_{{n_0}}} = + \infty {\text { if }}{x_n} = x + {a_n} and x ∈ [ 0 , 1 − a n 0 ] x \in [0,1 - {a_{{n_0}}}] .

Typical continuous functions are virtually nonmonotone

For every porosity premeasure ϕ \phi , a typical continuous function meets every monotone function in a bilaterally strongly ϕ \phi -porous set. The statement does not remain valid if we replace the class of monotone functions by the class of absolutely continuous functions.

Porosity Characteristics of Intersection Sets with the Typical Continuous Function

Porosity Characteristics of Intersection Sets with the Typical Continuous Function