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7.6 Implications of Continuity 231 An immediate consequence of Exercise 7.3.2 is the following. Theorem 7.5.20 A function f is continuous at a if and only if it is both left and right continuous at a. In defining continuity on a set, Definition 7.5.17 stipulated that the set must be open. This allowed for a definition of continuity on (a, b), but not on [a, b]. One-sided continuity now allows us to define continuity on [a, b] in such a way that we do not concern ourselves with how f behaves or whether it even exists outside of [a, b]. Definition 7.5.21 A function f is said to be continuous on [a, b] provided the following hold: 1. f is continuous on (a, b), 2. f is right continuous at a, 3. f is left continuous at b. Definition 7.5.21 can be naturally adapted to apply to the following example by omitting stipulation 3. Example 7.5.22 f(x) = √ x is continuous on [0, +∞). From Exercise 7.5.18, f is continuous at all positive real numbers, and by Exercise 7.3.5, it is right continuous at 0. 7.6 Implications of Continuity Continuity has many implications, and in this section we look at three of them. The second of these results is actually logically equivalent to continuity. 7.6.1 The Intermediate Value Theorem The imagery that we can sketch a continuous function without picking up the pencil makes this first result seem plausible. It says that a continuous function cannot be negative at one point and positive somewhere else without crossing the x-axis somewhere between the two points. One case of this is stated in the next exercise. To prove it, imagine standing on the x-axis at a and looking up the x-axis. There is a natural subset of the real numbers that is both non-empty and bounded from above by b to which the LUB property can apply. EXERCISE 7.6.1 Suppose a
232 Chapter 7 Functions of a Real Variable If g is a continuous function such that g(a) > 0 >g(b)for some a 0 and then for the case a2n+1 < 0. (a) Use your definitions from Exercise 7.4.9 to prove lim x =+∞ and lim x =−∞ (7.30) x→+∞ x→−∞ (b) With part (a) in hand, a result like Exercise 7.4.11 would be demonstrable for x →±∞by paralleling its proof. Assuming this result, show that P(x) as defined above and with a2n+1 > 0 satisfies lim P(x) =−∞ and lim P(x) =+∞ (7.31) x→−∞ x→+∞ 3 Consider the function g(x) = f(x) − x. If either g(a) or g(b) is zero, you’re done. Otherwise, apply the IVT.
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
Page 204 and 205: 5.6 Compactness 181 EXERCISE 5.6.3
Page 206 and 207: 5.6 Compactness 183 EXERCISE 5.6.13
Page 208 and 209: Sequences of Real Numbers 6.1 Seque
Page 210 and 211: 6.1 Sequences Defined 187 Example 6
Page 212 and 213: EXERCISE 6.1.15 Consider the sequen
Page 214 and 215: L 1 e L L 2 e . . . . . . 1 2 3 4 N
Page 216 and 217: 6.2 Convergence of Sequences 193 EX
Page 218 and 219: 6.2 Convergence of Sequences 195 To
Page 220 and 221: 6.3 The Nested Interval Property 19
Page 226 and 227: a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
Page 228 and 229: Example 6.4.7 The terms a0 = 1 a1 =
Page 230 and 231: Functions of a Real Variable 7 Func
Page 232 and 233: 7.1 Bounded and Monotone Functions
Page 234 and 235: 50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
Page 236 and 237: 7.2 Limits and Their Basic Properti
Page 238 and 239: 7.2 Limits and Their Basic Properti
Page 240 and 241: 7.3 More on Limits 217 values of 0
Page 242 and 243: 7.4 Limits Involving Infinity 219 W
Page 244 and 245: 7.4 Limits Involving Infinity 221 T
Page 246 and 247: (a) limx→a f(x) =−∞ (b) limx
Page 248 and 249: 7.5 Continuity 225 If f is not cont
Page 250 and 251: 1 2 3 4 5 6 Figure 7.6 Some example
Page 252 and 253: |a − x| |xa| = |x − a| |x||a| 7
Page 256 and 257: 7.6 Implications of Continuity 233
Page 258 and 259: 7.7 Uniform Continuity 235 we decla
Page 260 and 261: 7.7 Uniform Continuity 237 Next, le
Page 262 and 263: 7.7.2 Uniform Continuity and Compac
Page 264 and 265: P A R T I I I Basic Principles of A
Page 266 and 267: Groups 8 In its simplest terms, alg
Page 268 and 269: 8.1 Introduction to Groups 245 are
Page 270 and 271: 8.1 Introduction to Groups 247 Defi
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Page 274 and 275: Show that C × is an abelian group
Page 276 and 277: 8.2 Subgroups 253 G1 and G3 are aut
Page 278 and 279: 8.2 Subgroups 255 Notice how proper
Page 280 and 281: 8.2 Subgroups 257 Example 8.2.15 su
Page 282 and 283: EXERCISE 8.2.23 If G is a group and
Page 284 and 285: 8.3 Quotient Groups 261 [0] ={...,
Page 286 and 287: 8.3 Quotient Groups 263 is standard
Page 288 and 289: Step 2: Define an operation ∗H on
Page 290 and 291: (iii) (Z + 3.8) +Z (Z + 1.2) (iv)
Page 292 and 293: then for a given f ∈ S and any a
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Page 296 and 297: 2 3 1 3 1 Figure 8.1 Rigid square u
Page 298 and 299: EXERCISE 8.4.12 Complete Table 8.64
Page 300 and 301: 8.5 Normal Subgroups 277 Theorem 8.
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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9.1 Rings and Fields 289 (R9) Multi
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9.1 Rings and Fields 291 we define
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9.2 Subrings 293 In addition to pro
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9.2 Subrings 295 p1 = q1 = 3. Verif
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9.3 Ring Properties 297 Theorem 9.3
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9.3 Ring Properties 299 EXERCISE 9.
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9.4 Ring Extensions 301 in the inte
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9.4 Ring Extensions 303 Example 9.4
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9.4 Ring Extensions 305 We insist t
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9.5 Ideals 307 An ideal, say a left
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9.6 Generated Ideals 309 Proof. Sup
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EXERCISE 9.6.6 In the integers, wha
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9.7 Prime and Maximal Ideals 313 Ev
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9.8 Integral Domains 315 EXERCISE 9
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9.8 Integral Domains 317 Corollary
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9.9 Unique Factorization Domains 31
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9.10 Principal Ideal Domains 321 al
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9.10 Principal Ideal Domains 323 So
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9.11 Euclidean Domains 325 next exe
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9.11 Euclidean Domains 327 Exercise
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Proof. Let f and g be nonzero polyn
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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