6.2 Deeper Properties of Continuous Functions

Size: px
Start display at page:

Download "6.2 Deeper Properties of Continuous Functions"

Transcription

1 6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS Deeper Properties of Continuous Functions 6.2. Intermediate Value Theorem and Consequences When one studies a function, one is usually interested in the values the function can have. In other words, one is interested in the range of the function. More precisely, if f is a function with domain D, one tries to answer questions of the type:. What kind of a set is the range R of f? 2. Under which conditions is the range of f bounded? 3. Under which conditions is it closed? 4. Does f have a minimum? a maximum? You should recognize some of these questions. You studied them in Calculus I. We answer some of these questions in this section. We begin by stating and proving a very important theorem: the intermediate value theorem. Intuitively, this theorem says that the idea of a continuous function is that its graph can be drawn without lifting the pencil. In other words, its graph has no holes or breaks. Thus, it takes on all values between any two it achieves. More precisely: Theorem 447 (Intermediate Value Theorem) Suppose that f is continuous on the closed interval I = [a, b], let A = f (a), and B = f (b), A B. If C is a number between A and B, then there exists a number c in (a, b) such that f (c) = C. If in addition f is strictly monotone on I then c is unique. Proof. The theorem has two parts: existence of c and uniqueness of c. We prove each part separately. Existence of c The reader should draw a picture corresponding to the situation of the theorem and represent on the picture the various quantities involved in the proof. Assume all the hypotheses of the theorem are satisfied. If A B, then either A > B or A < B. We do the proof in the case A > B. The case A < B is identical, and is left to the reader. We look at g (x) = f (x) C. Since f is continuous on [a, b] and C is a constant, g is also continuous on [a, b]. Furthermore, by our assumption, g (a) > 0. Since g is continuous, by theorem 439, g will remain strictly positive to the right of a. For the same reason, g is strictly negative at b, and therefore to the left of b. Thus, the set S = {x I such that f (x) C > 0} is not empty. Furthermore, it is a subset of I and therefore is bounded. We saw in chapter that every non-empty bounded set had a supremum. Let c = sup S. The claim is that c is the number in the theorem, in other words, f (c) = C. We prove this fact by contradiction. Assume that f (c) C. Then, either f (c) > C or f (c) < C.

2 70 CHAPTER 6. CONTINUITY case : f (c) > C. By continuity of f, f (x) will remain strictly greater than C for a while. In other words, there is a number x > c such that f (x ) > C. Thus, x S and x > c. This contradicts the fact that c is an upper bound for S. case2: f (c) < C. Again, by continuity of f, f (x) < C to the left of c. Thus, there exists x < c such that f (x ) < C. Thus, x < c is an upper bound for S, which contradicts the fact that c = sup S. Therefore, f (c) = C Uniqueness of c We do a proof by contradiction. Assume c is not unique; that is there exists at least two such numbers, call them c and c 2. Without loss of generality, we may assume that c < c 2. f is either strictly increasing or strictly decreasing. case : f is strictly increasing. In this case, f (c ) < f (c 2 ). Therefore, we have: C = f (c ) < f (c 2 ) = C which is a contradiction. case 2: f is strictly decreasing. In this case, f (c ) > f (c 2 ). Therefore, we have: C = f (c ) > f (c 2 ) = C which is also a contradiction. This theorem is very important for both theoretical and practical reasons. On the theoretical side, as we will see shortly, many important results depend on this theorem. On the practical side, this theorem can be used to prove that certain equations have solutions. It can also be used to find approximations of these solutions. In fact, these approximations can be computed with as much accuracy as one needs. We illustrate this by an example. Example 448 Show that x 4 + 0x 3 207x 2 70x = 0 has a solution between 2 and 3. Find an approximation of this solution correct to 2 decimal places. Let f (x) = x 4 + 0x 3 207x 2 70x Then, f is continuous for all real numbers, so it is continuous on [2, 3]. In addition, f (2) = 528 and f (3) = 322. So, 0 is between f (2) and f (3). By the intermediate value theorem, there exists a number c between 2 and 3 such that f (c) = 0. To find an approximation, we compute f (2.), f (2.2),..., f (2.9) until we find two consecutive values for which f changes sign. In this case, f (2.6) = and f (2.7) = So, repeating the same argument as above, we find that c is between 2.6 and 2.7. We repeat the same procedure with f (2.6), f (2.62),..., f (2.69). We find that f (2.64) = and f (2.65) = By the same argument as above, c is between 2.64 and So, we know that the 2.6 part is correct. We need

3 6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 7 to repeat the procedure one more time since we want the solution correct to 2 decimal places. We compute f (2.64), f (2.642),..., f (2.649). We find that f (2.645) = and f (2.646) =.297. Therefore, c is between and Thus, 2.64 is part of the exact answer. Corollary 449 Let f be continuous on an interval I (any interval). Then, the range of f is also an interval. Proof. See homework. This theorem tells us that if the domain of f consists of one piece, so will its range. Some of the problems below as the reader to sketch some graphs relating a function and its range. The reader should do them before reading further, they will help in the understanding of these concepts. Note in particular that the theorem does not say that the range and the domain are the same type of intervals. One could be closed, the other could be open. One could be bounded, the other not. In the previous theorem, I was any interval. When we study f on a closed interval, we can reach even more powerful conclusions, as the next theorems show. Corollary 450 Let f be continuous on a closed interval [a, b]. Then, the range of f is bounded. Proof. See homework. Theorem 45 Let f be defined and continuous on [a, b]. Then, f attains the supremum and the infimum of its range. Proof. We show that f attains the supremum of its range. Let R denote the range of f. By the previous theorem, we know R is bounded, thus bounded above. Since it is not empty (f (a) R), it has a supremum. Let C = sup R. We need to show there exists c [a, b] such that f (c) = C. Define g (x) = C f (x) for x [a, b]. Then, either g (x) = 0 for some x in [a, b] or g (x) > 0 for all x in [a, b]. case : g (x) = 0 for some x in [a, b]. Then, we are done. case 2: g (x) > 0 for all x in [a, b]. We show this cannot happen. Suppose it did. Then, g would be continuous and non zero on [a, b] (why?). Therefore, the function would also be continuous on [a, b] (why?). Thus, by the g (x) previous theorem, would be bounded on [a, b]. Because C = sup R, g given ɛ > 0, one can find x [a, b] such that 0 < g (x) < ɛ (why?), thus g (x) > ɛ. It follows that cannot be bounded, which is a contradiction. g Corollary 452 Let f be defined and continuous on [a, b], then the range of f is a closed interval. Proof. See exercises at the end of the section

4 72 CHAPTER 6. CONTINUITY Corollary 453 Let f be defined and continuous on [a, b]. Then, there exists x and x 2 in [a, b] such that for all x [a, b], we have: f (x ) f (x) f (x 2 ) Proof. See exercises at the end of the section We have seen some interesting properties of continuous functions. First, we saw that if I is an interval and f is a continuous function, then f (I) is also an interval. Thus, a continuous function preserves intervals. We also saw that the image of a closed interval was a closed interval. More precisely, f ([a, b]) = [m, M] where m = inf (f ([a, b])), M = sup (f ([a, b])). We might ask ourselves what other properties of sets are preserved by continuous functions. These are important questions that have been asked by mathematicians. If you want to know some of these answers, keep taking math courses! Topological Interpretation of Continuity If you had taken a topology class before real analysis, you would have seen a different definition of continuity. What we list below as a theorem is taken as the topological definition of continuity. We show the two definitions, though very different, are equivalent. In proving the theorem, we will use our definition of continuity. Theorem 454 Let E be a subset of R and f : E R a function. Then: f continuous f (V ) open in E for every open subset V of R Proof. We outline the proof and leave the details as homework. Suppose f is continuous on E. Let V R be open, show that f (V ) is open. Pick a f (V ), show that a is an interior point that is there exists δ > 0 such that (a δ, a + δ) f (V ). But f (a) V which is open, so there exists ɛ > 0 such that (f (a) ɛ, f (a) + ɛ) V. Since f is continuous, for each ɛ, there exists a δ. See how the δ you get from continuity can be used to show (a δ, a + δ) f (V ). Suppose f (V ) open in E for every open subset V of R, show f is continuous on E. We need to show f is continuous at every point of E. The steps are similar to those of the other direction Exercises. Prove that x4 + 2x x 7. + x6 + 2x x 7 = 0 has a solution between and

5 6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS Same question as the previous problem for g (x) = 5 x + 7 x x 3 between and Prove corollary Prove corollary In the proof of theorem 45 there are three statements you were asked to justify. They are the statements with (why?). Justify these statements. 6. Prove corollary Prove corollary Sketch the graph of a continuous function on an open interval whose range is also an open interval. 9. Sketch the graph of a continuous function on an open interval whose range is a closed interval. 0. Sketch the graph of a function on an interval of the form [a, ) whose range is a closed interval.. Illustrate with an example the fact that if f is not continuous at a point, then its range may not be an interval, even if its domain is a closed interval. 2. Finish proving theorem 454.

5.5 Deeper Properties of Continuous Functions

5.5 Deeper Properties of Continuous Functions 5.5. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 195 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested

More information

5.5 Deeper Properties of Continuous Functions

5.5 Deeper Properties of Continuous Functions 200 CHAPTER 5. LIMIT AND CONTINUITY OF A FUNCTION 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested

More information

2.2 Some Consequences of the Completeness Axiom

2.2 Some Consequences of the Completeness Axiom 60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that

More information

Consequences of the Completeness Property

Consequences of the Completeness Property Consequences of the Completeness Property Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of the Completeness Property Today 1 / 10 Introduction In this section, we use the fact that R

More information

6.2 Important Theorems

6.2 Important Theorems 6.2. IMPORTANT THEOREMS 223 6.2 Important Theorems 6.2.1 Local Extrema and Fermat s Theorem Definition 6.2.1 (local extrema) Let f : I R with c I. 1. f has a local maximum at c if there is a neighborhood

More information

5.4 Continuity: Preliminary Notions

5.4 Continuity: Preliminary Notions 5.4. CONTINUITY: PRELIMINARY NOTIONS 181 5.4 Continuity: Preliminary Notions 5.4.1 Definitions The American Heritage Dictionary of the English Language defines continuity as an uninterrupted succession,

More information

106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S)

106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S) 106 CHAPTER 3. TOPOLOGY OF THE REAL LINE 3.3 Limit Points 3.3.1 Main Definitions Intuitively speaking, a limit point of a set S in a space X is a point of X which can be approximated by points of S other

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

Solution of the 7 th Homework

Solution of the 7 th Homework Solution of the 7 th Homework Sangchul Lee December 3, 2014 1 Preliminary In this section we deal with some facts that are relevant to our problems but can be coped with only previous materials. 1.1 Maximum

More information

Math 104: Homework 7 solutions

Math 104: Homework 7 solutions Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for

More information

Upper and Lower Bounds

Upper and Lower Bounds James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 30, 2017 Outline 1 2 s 3 Basic Results 4 Homework Let S be a set of real numbers. We

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty.

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. 1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. Let E be a subset of R. We say that E is bounded above if there exists a real number U such that x U for

More information

Metric spaces and metrizability

Metric spaces and metrizability 1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively

More information

Differentiation - Important Theorems

Differentiation - Important Theorems Differentiation - Important Theorems Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Differentiation - Important Theorems Spring 2012 1 / 10 Introduction We study several important theorems related

More information

MATH CALCULUS I 1.5: Continuity

MATH CALCULUS I 1.5: Continuity MATH 12002 - CALCULUS I 1.5: Continuity Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 12 Definition of Continuity Intuitively,

More information

Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases

Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases September 22, 2018 Recall from last week that the purpose of a proof

More information

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ). Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.

More information

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More information

Math 320-2: Midterm 2 Practice Solutions Northwestern University, Winter 2015

Math 320-2: Midterm 2 Practice Solutions Northwestern University, Winter 2015 Math 30-: Midterm Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A metric on R with respect to which R is bounded. (b)

More information

A lower bound for X is an element z F such that

A lower bound for X is an element z F such that Math 316, Intro to Analysis Completeness. Definition 1 (Upper bounds). Let F be an ordered field. For a subset X F an upper bound for X is an element y F such that A lower bound for X is an element z F

More information

Introduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION

Introduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from

More information

Important Properties of R

Important Properties of R Chapter 2 Important Properties of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference

More information

a. Do you think the function is linear or non-linear? Explain using what you know about powers of variables.

a. Do you think the function is linear or non-linear? Explain using what you know about powers of variables. 8.5.8 Lesson Date: Graphs of Non-Linear Functions Student Objectives I can examine the average rate of change for non-linear functions and learn that they do not have a constant rate of change. I can determine

More information

Consequences of Continuity

Consequences of Continuity Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The

More information

A lower bound for X is an element z F such that

A lower bound for X is an element z F such that Math 316, Intro to Analysis Completeness. Definition 1 (Upper bounds). Let F be an ordered field. For a subset X F an upper bound for X is an element y F such that A lower bound for X is an element z F

More information

Week 2: Sequences and Series

Week 2: Sequences and Series QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime

More information

2.4 The Extreme Value Theorem and Some of its Consequences

2.4 The Extreme Value Theorem and Some of its Consequences 2.4 The Extreme Value Theorem and Some of its Consequences The Extreme Value Theorem deals with the question of when we can be sure that for a given function f, (1) the values f (x) don t get too big or

More information

MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7

MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 Real Number Summary of terminology and theorems: Definition: (Supremum & infimum) A supremum (or least upper bound) of a non-empty

More information

Real Variables: Solutions to Homework 3

Real Variables: Solutions to Homework 3 Real Variables: Solutions to Homework 3 September 3, 011 Exercise 0.1. Chapter 3, # : Show that the cantor set C consists of all x such that x has some triadic expansion for which every is either 0 or.

More information

From Calculus II: An infinite series is an expression of the form

From Calculus II: An infinite series is an expression of the form MATH 3333 INTERMEDIATE ANALYSIS BLECHER NOTES 75 8. Infinite series of numbers From Calculus II: An infinite series is an expression of the form = a m + a m+ + a m+2 + ( ) Let us call this expression (*).

More information

The Real Number System

The Real Number System MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely

More information

Lecture 2. Econ August 11

Lecture 2. Econ August 11 Lecture 2 Econ 2001 2015 August 11 Lecture 2 Outline 1 Fields 2 Vector Spaces 3 Real Numbers 4 Sup and Inf, Max and Min 5 Intermediate Value Theorem Announcements: - Friday s exam will be at 3pm, in WWPH

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Chapter One. The Real Number System

Chapter One. The Real Number System Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and

More information

MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE

MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE SEBASTIEN VASEY These notes describe the material for November 26, 2018 (while similar content is in Abbott s book, the presentation here is different).

More information

Structure of R. Chapter Algebraic and Order Properties of R

Structure of R. Chapter Algebraic and Order Properties of R Chapter Structure of R We will re-assemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions

More information

Fixed-Point Iteration

Fixed-Point Iteration Fixed-Point Iteration MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013 Motivation Several root-finding algorithms operate by repeatedly evaluating a function until its

More information

Consequences of Continuity

Consequences of Continuity Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline Domains of Continuous Functions The Intermediate

More information

Describing the Real Numbers

Describing the Real Numbers Describing the Real Numbers Anthony Várilly Math 25a, Fall 2001 1 Introduction The goal of these notes is to uniquely describe the real numbers by taking certain statements as axioms. This exercise might

More information

Scalar multiplication and addition of sequences 9

Scalar multiplication and addition of sequences 9 8 Sequences 1.2.7. Proposition. Every subsequence of a convergent sequence (a n ) n N converges to lim n a n. Proof. If (a nk ) k N is a subsequence of (a n ) n N, then n k k for every k. Hence if ε >

More information

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement. Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

In N we can do addition, but in order to do subtraction we need to extend N to the integers Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need

More information

2.3 Some Properties of Continuous Functions

2.3 Some Properties of Continuous Functions 2.3 Some Properties of Continuous Functions In this section we look at some properties, some quite deep, shared by all continuous functions. They are known as the following: 1. Preservation of sign property

More information

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1. Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

More information

Relationship Between Integration and Differentiation

Relationship Between Integration and Differentiation Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16 Introduction In the previous sections we defined

More information

Homework 4, 5, 6 Solutions. > 0, and so a n 0 = n + 1 n = ( n+1 n)( n+1+ n) 1 if n is odd 1/n if n is even diverges.

Homework 4, 5, 6 Solutions. > 0, and so a n 0 = n + 1 n = ( n+1 n)( n+1+ n) 1 if n is odd 1/n if n is even diverges. 2..2(a) lim a n = 0. Homework 4, 5, 6 Solutions Proof. Let ɛ > 0. Then for n n = 2+ 2ɛ we have 2n 3 4+ ɛ 3 > ɛ > 0, so 0 < 2n 3 < ɛ, and thus a n 0 = 2n 3 < ɛ. 2..2(g) lim ( n + n) = 0. Proof. Let ɛ >

More information

The definition, and some continuity laws. Types of discontinuities. The Squeeze Theorem. Two special limits. The IVT and EVT.

The definition, and some continuity laws. Types of discontinuities. The Squeeze Theorem. Two special limits. The IVT and EVT. MAT137 - Week 5 The deadline to drop to MAT135 is tomorrow. (Details on course website.) The deadline to let us know you have a scheduling conflict with Test 1 is also tomorrow. (Details on the course

More information

INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES

INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES You will be expected to reread and digest these typed notes after class, line by line, trying to follow why the line is true, for example how it

More information

Studying Rudin s Principles of Mathematical Analysis Through Questions. August 4, 2008

Studying Rudin s Principles of Mathematical Analysis Through Questions. August 4, 2008 Studying Rudin s Principles of Mathematical Analysis Through Questions Mesut B. Çakır c August 4, 2008 ii Contents 1 The Real and Complex Number Systems 3 1.1 Introduction............................................

More information

Walker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015

Walker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015 Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of

More information

Set, functions and Euclidean space. Seungjin Han

Set, functions and Euclidean space. Seungjin Han Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,

More information

MA103 Introduction to Abstract Mathematics Second part, Analysis and Algebra

MA103 Introduction to Abstract Mathematics Second part, Analysis and Algebra 206/7 MA03 Introduction to Abstract Mathematics Second part, Analysis and Algebra Amol Sasane Revised by Jozef Skokan, Konrad Swanepoel, and Graham Brightwell Copyright c London School of Economics 206

More information

Problem Set 2: Solutions Math 201A: Fall 2016

Problem Set 2: Solutions Math 201A: Fall 2016 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that

More information

M17 MAT25-21 HOMEWORK 6

M17 MAT25-21 HOMEWORK 6 M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute

More information

MATH 131A: REAL ANALYSIS (BIG IDEAS)

MATH 131A: REAL ANALYSIS (BIG IDEAS) MATH 131A: REAL ANALYSIS (BIG IDEAS) Theorem 1 (The Triangle Inequality). For all x, y R we have x + y x + y. Proposition 2 (The Archimedean property). For each x R there exists an n N such that n > x.

More information

MATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals

MATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals MATH 02 INTRODUCTION TO MATHEMATICAL ANALYSIS Properties of Real Numbers Some Fundamentals The whole course will be based entirely on the study of sequence of numbers and functions defined on the real

More information

MATH 117 LECTURE NOTES

MATH 117 LECTURE NOTES MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set

More information

Part 2 Continuous functions and their properties

Part 2 Continuous functions and their properties Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice

More information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.2 Polynomial Functions of Higher Degree Copyright Cengage Learning. All rights reserved. What You Should Learn Use

More information

Lower semicontinuous and Convex Functions

Lower semicontinuous and Convex Functions Lower semicontinuous and Convex Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2017 Outline Lower Semicontinuous Functions

More information

Supremum and Infimum

Supremum and Infimum Supremum and Infimum UBC M0 Lecture Notes by Philip D. Loewen The Real Number System. Work hard to construct from the axioms a set R with special elements O and I, and a subset P R, and mappings A: R R

More information

Solutions Final Exam May. 14, 2014

Solutions Final Exam May. 14, 2014 Solutions Final Exam May. 14, 2014 1. (a) (10 points) State the formal definition of a Cauchy sequence of real numbers. A sequence, {a n } n N, of real numbers, is Cauchy if and only if for every ɛ > 0,

More information

CHAPTER 8: EXPLORING R

CHAPTER 8: EXPLORING R CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed

More information

Solution of the 8 th Homework

Solution of the 8 th Homework Solution of the 8 th Homework Sangchul Lee December 8, 2014 1 Preinary 1.1 A simple remark on continuity The following is a very simple and trivial observation. But still this saves a lot of words in actual

More information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined

More information

1 Homework. Recommended Reading:

1 Homework. Recommended Reading: Analysis MT43C Notes/Problems/Homework Recommended Reading: R. G. Bartle, D. R. Sherbert Introduction to real analysis, principal reference M. Spivak Calculus W. Rudin Principles of mathematical analysis

More information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #1. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #1. Solutions Math 501 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 015 EXERCISES FROM CHAPTER 1 Homework #1 Solutions The following version of the

More information

Definition (The carefully thought-out calculus version based on limits).

Definition (The carefully thought-out calculus version based on limits). 4.1. Continuity and Graphs Definition 4.1.1 (Intuitive idea used in algebra based on graphing). A function, f, is continuous on the interval (a, b) if the graph of y = f(x) can be drawn over the interval

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

In N we can do addition, but in order to do subtraction we need to extend N to the integers Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend

More information

Math 328 Course Notes

Math 328 Course Notes Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the

More information

V. Graph Sketching and Max-Min Problems

V. Graph Sketching and Max-Min Problems V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.

More information

Notes on the Point-Set Topology of R Northwestern University, Fall 2014

Notes on the Point-Set Topology of R Northwestern University, Fall 2014 Notes on the Point-Set Topology of R Northwestern University, Fall 2014 These notes give an introduction to the notions of open and closed subsets of R, which belong to the subject known as point-set topology.

More information

Math 140: Foundations of Real Analysis. Todd Kemp

Math 140: Foundations of Real Analysis. Todd Kemp Math 140: Foundations of Real Analysis Todd Kemp Contents Part 1. Math 140A 5 Chapter 1. Ordered Sets, Ordered Fields, and Completeness 7 1. Lecture 1: January 5, 2016 7 2. Lecture 2: January 7, 2016

More information

Hence, (f(x) f(x 0 )) 2 + (g(x) g(x 0 )) 2 < ɛ

Hence, (f(x) f(x 0 )) 2 + (g(x) g(x 0 )) 2 < ɛ Matthew Straughn Math 402 Homework 5 Homework 5 (p. 429) 13.3.5, 13.3.6 (p. 432) 13.4.1, 13.4.2, 13.4.7*, 13.4.9 (p. 448-449) 14.2.1, 14.2.2 Exercise 13.3.5. Let (X, d X ) be a metric space, and let f

More information

Math 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1

Math 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1 Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,

More information

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3 Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability

More information

POL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005

POL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005 POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1

More information

3 Lecture Separation

3 Lecture Separation 3 Lecture 3 3.1 Separation We are now going to move on to an extremely useful set of results called separation theorems. Basically, these theorems tell us that, if we have two convex sets, then we can

More information

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k)

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k) MATH 111 Optional Exam 3 lutions 1. (0 pts) We define a relation on Z as follows: a b if a + b is divisible by 3. (a) (1 pts) Prove that is an equivalence relation. (b) (8 pts) Determine all equivalence

More information

2.1 Convergence of Sequences

2.1 Convergence of Sequences Chapter 2 Sequences 2. Convergence of Sequences A sequence is a function f : N R. We write f) = a, f2) = a 2, and in general fn) = a n. We usually identify the sequence with the range of f, which is written

More information

Due date: Monday, February 6, 2017.

Due date: Monday, February 6, 2017. Modern Analysis Homework 3 Solutions Due date: Monday, February 6, 2017. 1. If A R define A = {x R : x A}. Let A be a nonempty set of real numbers, assume A is bounded above. Prove that A is bounded below

More information

1 The Real Number System

1 The Real Number System 1 The Real Number System The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth century, as a kind of an envelope

More information

Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization

Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Compiled by David Rosenberg Abstract Boyd and Vandenberghe s Convex Optimization book is very well-written and a pleasure to read. The

More information

2MA105 Algebraic Structures I

2MA105 Algebraic Structures I 2MA105 Algebraic Structures I Per-Anders Svensson http://homepage.lnu.se/staff/psvmsi/2ma105.html Lecture 12 Partially Ordered Sets Lattices Bounded Lattices Distributive Lattices Complemented Lattices

More information

Indeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( )

Indeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( ) Lebesgue Measure The idea of the Lebesgue integral is to first define a measure on subsets of R. That is, we wish to assign a number m(s to each subset S of R, representing the total length that S takes

More information

Continuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics

Continuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics Continuity MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Fall 2017 Intuitive Idea A process or an item can be described as continuous if it exists without interruption. The mathematical

More information

HOMEWORK ASSIGNMENT 6

HOMEWORK ASSIGNMENT 6 HOMEWORK ASSIGNMENT 6 DUE 15 MARCH, 2016 1) Suppose f, g : A R are uniformly continuous on A. Show that f + g is uniformly continuous on A. Solution First we note: In order to show that f + g is uniformly

More information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

Uniform Convergence Examples

Uniform Convergence Examples Uniform Convergence Examples James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 13, 2017 Outline 1 Example Let (x n ) be the sequence

More information

Metric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)

Metric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.

More information

4130 HOMEWORK 4. , a 2

4130 HOMEWORK 4. , a 2 4130 HOMEWORK 4 Due Tuesday March 2 (1) Let N N denote the set of all sequences of natural numbers. That is, N N = {(a 1, a 2, a 3,...) : a i N}. Show that N N = P(N). We use the Schröder-Bernstein Theorem.

More information

PERIODIC POINTS OF THE FAMILY OF TENT MAPS

PERIODIC POINTS OF THE FAMILY OF TENT MAPS PERIODIC POINTS OF THE FAMILY OF TENT MAPS ROBERTO HASFURA-B. AND PHILLIP LYNCH 1. INTRODUCTION. Of interest in this article is the dynamical behavior of the one-parameter family of maps T (x) = (1/2 x

More information

Solutions to Homework 9

Solutions to Homework 9 Solutions to Homework 9 Read the proof of proposition 1.7 on p. 271 (section 7.1). Write a more detailed proof. In particular, state the defintion of uniformly continuous and explain the comment whose

More information

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA  address: Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework

More information

g 2 (x) (1/3)M 1 = (1/3)(2/3)M.

g 2 (x) (1/3)M 1 = (1/3)(2/3)M. COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is

More information

Math General Topology Fall 2012 Homework 11 Solutions

Math General Topology Fall 2012 Homework 11 Solutions Math 535 - General Topology Fall 2012 Homework 11 Solutions Problem 1. Let X be a topological space. a. Show that the following properties of a subset A X are equivalent. 1. The closure of A in X has empty

More information

Mathematical Induction Again

Mathematical Induction Again Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Mathematical Induction Simple POMI Examples

More information

Rational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions

Rational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W

More information

1 The Glivenko-Cantelli Theorem

1 The Glivenko-Cantelli Theorem 1 The Glivenko-Cantelli Theorem Let X i, i = 1,..., n be an i.i.d. sequence of random variables with distribution function F on R. The empirical distribution function is the function of x defined by ˆF

More information