# Counterexamples on uniform convergence pdf

Download counterexamples in calculus or read online books in pdf, epub, tuebl, and mobi format. A comprehensive and thorough analysis of concepts and results on uniform convergence. We look here at the continuity of a sequence of functions that converges pointwise and give some counterexamples of what happens versus uniform convergence recalling the definition of pointwise convergence. Does the fourier series of a continuous function on the circle converge to it pointwise. Math 401 notes sequences of functions pointwise and. Real analysis michael boardman, pacific universitychair. Sequences of functions pointwise and uniform convergence fall 2005 previously, we have studied sequences of real numbers. In section 1 pointwise and uniform convergence of sequences of functions are discussed and examples are given. Uniform convergence in this section, we introduce a stronger notion of convergence of functions than pointwise convergence, called uniform convergence.

Sequences, series, functions, and integrals free ebook download authors. This website gives you access to the rich tools and resources available for this text. A sequence of functions f n is a list of functions f 1,f 2. We formulate and prove theorems for the positive results, and provide counterexamples for the negative. Pointwise and uniform convergence of sequences and. How do we determine whether convergence is uniform in particular cases. Sequences of functions pointwise and uniform convergence i. Which of the following sequences of functions converge uniformly on x. The 1st half of the book discusses the real number system, functions and limits, differentiation, riemann integration, sequences, infinite series, more.

We will also see that uniform convergence is what allows us to integrate and di. Ludmila bourchtein this book presents counterexamples to false statements typically found within the study of mathematical analysis, real analysis, and calculus, all of which are related to infinite sequences, series. Chapter 9 uniform convergence, integration and power series. This sequence does not converge pointwise on r because lim n. In most cases, this is done along the lines of one of the two following remarks.

We consider here real functions defined on a closed interval \a,b\. Counterexamples related to the main results follow, along with motivation for questions and counterstatements that range in difficulty. The development of exceptions into counterexamples is consequently explained through the evolution of formulacentered mathematics into. In section 2 the three theorems on exchange of pointwise limits, inte gration and di erentiation which are corner stones for all later development are proven.

Now we discuss the topic of sequences of real valued functions. Therefore, uniform convergence implies pointwise convergence. Uniform convergence uniform convergence is the main theme of this chapter. In probability theory, there exist several different notions of convergence of random variables. The book includes the convergence of sequences, series and families of. Example 10 let fn be the sequence of functions on 0. The book includes the convergence of sequences, series and families of functions, and proper and improper integrals depending on a parameter.

This web site gives you access to the rich tools and resources available for this text. Counterexamples to a continuous function preserving almost uniform convergence and convergence in measure. Pdf download counterexamples in topology dover books on. In the mathematical field of analysis, uniform convergence is a mode of convergence of. Sequences, series, functions, and integrals presents counterexamples to false statements typically found within the study of mathematical analysis and calculus, all of which are related to uniform convergence. This chapter continues the study of the consequence of uniform convergence of a series of function. In is a sequence of continuous functions on the compact metric space k that converges pointwise to. The book includes the convergence of sequences, series and families of functions, and proper and improper integrals depending.

In particular, uniform convergence may seem even more remote, and therefore what id like to do now issaving the formal proofs for the supplementary notes, let me show you pictorially just what the concept of uniform convergence really is. But the converse is false as we can see from the following counterexample. Sequences of functions pointwise and uniform convergence. Mathematics students as well as students of engineering and science will find here a superb treatment, carefully thought out and well presented, that is ideal for a one semester course. Sequences, series, functions, and integrals presents counterexamples to false statements typically found within the stud. Real analysis nu math sites northwestern university. Welcome to the instructor companion site for counterexamples on uniform convergence. Chapter 9 uniform convergence, integration and power series recall that theorem 7 of chapter 8 says that if a sequence of continuous functions gn converges uniformly on a, b to a function g, then the integral of the limiting function g is the limit. The interval of uniform convergence is always to be a closed interval, that is it must include the end points. Sequences, series, functions, and integrals is an appropriate reference andor supplementary reading for upperundergraduate and graduatelevel courses in mathematical analysis and advanced calculus for students majoring in mathematics, engineering, and other sciences.

Section 6 contains some illustrative counterexamples concerning uniform laws of large numbers and the relationship of the respective equicontinuitytype concepts. Pointwise convergence not uniform on any interval january 8, 2017 jeanpierre merx leave a comment we provide in this article an example of a pointwise convergent sequence of real functions that doesnt converge uniformly on any interval. Click download or read online button to get counterexamples in calculus book now. In chapter 7 we have observed that the uniform limit of a sequence of continuous function is continuous theorem 14 chapter 7. Then by the limit formula for the exponential, which we do not prove here, fn ex pointwise on r. These counterexamples deal mostly with the part of analysis known as real variables. Generic uniform convergence and equicontinuity concepts. Save up to 80% by choosing the etextbook option for isbn. If fn is pointwise convergent then the function defined by fx lim n. Erratum to uniform convergence and chaotic behavior. A comprehensive and thorough analysis of concepts and results on uniform convergence counterexamples on uniform convergence.

The same concepts are known in more general mathematics as stochastic convergence and they. The 2nd half examines functions of 2 variables, plane sets, area, metric and topological spaces, and function spaces. Prove there exist plenty of counterexamples using the uniform. In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. It was against this background of cauchys proof of the. Three counterexamples to recent results on finite and fixedtime convergent controllers and observers article pdf available in automatica february. A familiarity with basic notions and results in probability and stochastic processes is assumed, and a chapter of supplementary remarks provides a wealth of information about original sources as well as. Sequences, series, functions, and integrals 1st edition by andrei bourchtein. Pondering examples and counterexamples for uniform convergence. Counterexamples in calculus download ebook pdf, epub.

That means i should have counterexamples for the following. Pdf three counterexamples to recent results on finite. In chapter 3 we consider the less common quasi uniform convergence in more detail, exploring the motivation for, and properties of, this convergence. Finally in chapter 4 the relations between the di erent types of convergence are examined. Sequences, series, functions, and integrals by andrei bourchtein. In chapter 7 we have observed that the uniform limit of a sequence of continuous function is.

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