## Recursos de colección

1. #### On the Derivatives of Functions of Bounded Variation

Cater, F. S.
ss{raeq} \usepackage{amsmath,amssymb,amsthm} %\coverauthor{F. S. Cater} %\covertitle{On the Derivatives of Functions of Bounded Variation} \received{November 5, 2000} \firstpagenumber{975} \markboth{F. S. Cater}{Questions on Derivatives of Functions of Bounded Variation} \author{F. S. Cater, Department of Mathematics, Portland State University, Portland, Oregon 97207, USA} \title{ON THE DERIVATIVES OF FUNCTIONS OF BOUNDED VARIATION} \begin{document} \maketitle The following two questions were submitted by F. S. Cater. Let $\cal F$ denote the family of all absolutely continuous , nondecreasing functions on $[0,1]$. Endow $\cal F$ withe the complete metric $d$ defined by $$d(f,g) = |f(0)-g(0)| + \int_o^1 |f'-g'|$$ Let \begin{align*} {\cal G}=\{g\in {\cal F};& g'(x) =...

2. #### The Persistence of ω-Limit Sets under Perturbation of the Generating Function

Steele, T. H.
We consider the set valued function $\Omega$ taking $f$ in $\ C(I,I)$ to its collection of $\omega$-limit sets $\Omega (f)=\{\omega (x,f):x\in I\}$, and consider how $\Omega (f)$ is affected by pertubations of $f$. Our main result characterizes those functions $f$ in $C(I,I)$ at which $\Omega$ is upper semicontinuous, so that whenever $g$ is sufficiently close to $f$, every $\omega$-limit set of $g$ is close to some $\omega$-limit set of $f$ in the Hausdorff metric space. We also develop necessary and sufficient conditions for a function $f$ in $C(I,I)$ to be a point of lower semicontinuity...

3. #### SAC Property and Approximate Semicontinuity

Prus-Wiśniowski, Franciszek; Szkibiel, Grzegorz
In this article we investigate approximate semicontinuity of a function related to Grande's SAC problem.

4. #### A Solution to Pfeffer’s Problem

Tuo-Yeong, Lee
We give an example of a non-integrable function $f$ on $[0,1]\times [0,1]$ such that $$\int_{\alpha}^{\beta} (\int_{\gamma}^{\delta} f(x,y)\,dy)\,dx =\int_{\gamma}^{\delta} (\int_{\alpha}^{\beta} f(x,y)\,dx)\,dy$$ for each subinterval $[\alpha,\beta] \times [\gamma, \delta]$ of $[0,1] \times [0,1]$.

5. #### On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions

Grande, Zbigniew
In this article we prove that if a function $f:X \to {\cal R}$ is the pointwise (discrete) [transfinite] limit of a sequence of real functions $f_n$ with closed graphs defined on complete separable metric space $X$ then $f$ is the pointwise (discrete) [transfinite] limit of a sequence of continuous functions. Moreover we show that each Lebesgue measurable function $f:{\cal R} \to {\cal R}$ is the discrete limit of a sequence of functions with closed graphs in the product topology $T_d\times T_e$, where $T_d$ denotes the density topology and $T_e$ the Euclidean topology.

6. #### On the Derivatives of Functions of Bounded Variation

Cater, F. S.
Using a standard complete metric $w$ on the set $F$ of continuous functions of bounded variation on the interval $[0,1]$, we find that a typical function in $F$ has an infinite derivative at continuum many points in every subinterval of $[0,1]$. Moreover, for a typical function in $F$, there are continuum many points in every subinterval of $[0,1]$ where it has no derivative, finite nor infinite. The restriction of the derivative of a typical function in $F$ to the set of points of differentiability has infinite oscillation at each point of this set.

7. #### Measure Zero Sets Whose Algebraic Sum Is Non-Measurable

Ciesielski, Krzysztof
In this note we will show that for every natural number $n>0$ there exists an $S\subset[0,1]$ such that its $n$-th algebraic sum $nS=S+\cdots +S$ is a nowhere dense measure zero set, but its $n+1$-st algebraic sum $nS+S$ is neither measurable nor it has the Baire property. In addition, the set $S$ will be also a Hamel base, that is, a linear base of $\mathbb{R}$ over $\mathbb{Q}$.

8. #### Functions I-Approximately Continuous in I1 A. E. Direction at Every Point

Szkopińska, Bożena
In this paper we shall show that every function $f\colon\Re^2\rightarrow \Re$ having the Baire property and ${\cal I}$-approximately continuous in ${\cal I}_1$-almost everywhere direction at every point is of the first class of Baire.

9. #### Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg

Ciesielski, Krzysztof; Pawlikowski, Janusz
We will show that the following set theoretical assumption \begin{quote} $\continuum=\omega_2$, the dominating number ${\mathfrak d}$ equals to $\omega_1$, and there exists an $\omega_1$-generated Ramsey ultrafilter on $\omega$ \end{quote} (which is consistent with ZFC) implies that for an arbitrary sequence $f_n\colon\real\to\real$ of uniformly bounded functions there is a set $P\subset\real$ of cardinality continuum and an infinite $W\subset\omega$ such that $\{f_n\restriction P\colon n\in W\}$ is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions $f_n$ are measurable or have the Baire property then $P$ can be chosen as a perfect set. We will also show that $\cof(\NN)=\omega_1$ implies...

10. #### A Remark on a Maximal Operator for Fourier Multipliers

Bourgain, J.; Kostyukovsky, S.; Olevskiǐ, A.
For a finite set $\Lambda$ on the circle we consider a family of the multiplier operators $T_m$ in $l_2 ({\mathbb Z})$ generated by the $2^{-m}$-neighborhoods of $\Lambda$. We show that the norm of the corresponding maximal operator $T$ can not be estimated by an absolute constant.

11. #### Graphs of Functions, Regular Sets and S-Straight Sets

Delaware, R.; Eifler, L.
A subset $E$ of $\Bbb{R}^p$ is s-straight if $E$ has finite Hausdorff s-dimensional outer measure which equals its Method I s-outer measure. The graph of a continuously differentiable function is shown to be the countable union of closed 1-straight sets together with a set of Hausdorff 1-measure zero. This result is extended to the graphs of absolutely continuous functions and to regular sets.

12. #### A Darboux Baire One Fixed Point Problem

Humke, P. D.; Svetic, R. E.; Weil, C. E.
K. Ciesielski asked whether the composition of two derivatives from the unit interval to itself always has a fixed point. The question is equivalent to asking if the composition of two Darboux, Baire one maps of $[0,1]$ to $[0,1]$ has a fixed point. The question is answered affirmatively for three subclasses of the Darboux, Baire one maps of $[0,1]$ to $[0,1]$

13. #### Some Examples of Meager Sets in Banach Spaces

Balcerzak, Marek; Wachowicz, Artur
We show that some sets in the spaces $c_0\times c_0$, $L^1[0,1]\times L^1[0,1]$ and $C[0,1]$, that appear in analysis, are meager.

14. #### A Fundamental Theorem of Calculus for the Kurzweil-Henstock Integral in ℝ m

Cabral, Emmanuel; Lee, Peng-Yee
In this paper, \ we \ give a characterization of the Kurzweil-Henstock integral in n-dimensional space.

15. #### Non-Uniqueness of Composition Square Roots

Dewsnap, D. J.; Fischer, P.
In response to a question posed by O.E. Lanford III, it is shown that for each $$\mu>0$$ there is a differentiable and non-linearizable interval map $g$ with non-vanishing derivative defined on a neighborhood of a fixed point $0$ with $$g^\prime(0)=\mu$$ such that $g$ has infinitely many differentiable and non-linearizable orientation-reversing composition square roots with non-vanishing first derivatives on a neighborhood of $0$.

16. #### Henstock-Stieltjes Integrals Not Induced by Measure

Pal, Suppriya; Ganguly, D. K.; Yee, Lee Peng
The present paper concerns with the introduction of a new type of generalized Stieltjes integral with an integrator function which depends on multiple points in a division and cannot be induced by a measure. Some properties of this integral were studied.

17. #### Packing Measure in General Metric Space

Edgar, G. A.
Packing measures are counterparts to Hausdorff measures, used in measuring fractal dimension of sets. C.~Tricot defined them for subsets of finite-di\-men\-sion\-al Euclidean space. We consider here the proper way to phrase the definitions for use in general metric spaces, and for Hausdorff functions other than the simple powers, in particular non-blanketed Hausdorff functions. The question of the Vitali property arises in this context. An example of a metric space due to R.~O. Davies illustrates the concepts.

18. #### Note on the Outer Measures of Images of Sets

Cater, F. S.
Let $f$ be a real function on ${\mathbb R}$, let $\{I_v\}$ be a family of intervals covering a set $E$ such that $m(E \cap I_v) \ge m\bigl (f(E \cap I_v)\bigr )$ for each $I_v$. We prove that $m\bigl (f(E)\bigr ) \le 2 \cdot m(E)$. No coefficient smaller than $2$ will suffice here in general.

19. #### Infinite Peano Derivatives

Laczkovich, Miklós
Let $f_{(n)}$ and ${\underline f}{_{(n)}}$ denote the $n^{\rm th}$ Peano derivative and the $n^{\rm th}$ lower Peano derivative of the function $f:[a,b]\to$\mathbb{R} .$We investigate the validity of the following statements. ¶$(M_n)$If the set$H=\{ x\in [a,b]: {\underline f}{_{(n)}} (x)>0\}$is of positive outer measure, then$f$is$n$-convex on a subset of$H$having positive outer measure. ¶$(Z_n)$The set$E_n (f)=\{ x\in [a,b] : f_{(n)} (x)=\infty \}$is of measure zero for every$f:[a,b]\to \mathbb{R} .$¶ We prove that ($M_n$) and ($Z_n$) are true for$n=1$and$n=2,$but false for$n\ge 3.$More precisely we show that for every$n\ge 3\$ there...

20. #### Asymptotics of the Quantization Errors for Self-Similar Probabilities

Graf, S.; Luschgy, H.
The formulae for determining the quantization dimensions of self--similar probabilities satisfying the open set condition are proved by a new method. In addition, this method gives the exact order of convergence for the quantization errors.

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