Index

A

Abel’s Theorem, 55

Algebra of functions, 137

“And” connective, 15

Applications:

    of compactness, 33–37

    of the Mean Value Theorem, 86–91

    of the Weierstrass M-test, 125–29

Arzela-Ascoli Theorem, 132–33

B

Baire Category Theorem, 144–49

    proof of, 145–46

Ball, 11

Bolzano-Weierstrass Theorem, 34–35

    corollary, 35–36

Boundary point, 23, 42

Bounded function, 65, 125

Bounded sets, 29–35, 161–62

Bounded sequence, 34

Bounded variation, functions of, 106–11

C

Calculus, fundamental theorem of, 104–5

Cantor diagonal process, 8

Cantor set, 76–80

Cantor’s Nested Set Property, 29, 164–65

Category 1 and category 2, 145

Cauchy product of power series, 139

Cauchy-Schwarz inequality, 52

    for integrals, 115

    for sums, 52

Cauchy sequence, 24, 40–41, 124, 164

Chain rule, 81–82

Change of variable formula, 112

Closed ball, 12

Closed sets, 12–13

    continuity and, 61–62

    relation between open sets and, 13–14

    unions and intersection of, 25–28

Closure, 22

Cluster point, 21

Co-finite set, 23

Codomain, 57

Compactness, 28–37, 161–64

    applications of, 33–37

    Bolzano-Weierstrass Theorem, 34–35

    and closed, bounded sets, 30

    and continuity, 64–70

    criteria for, 162

    definition, 30

    Heine-Borel Theorem, 31–32, 35, 130, 164

    nested set property, 29

Compact set, 30

Complement, 2, 3, 6

Complete metric space, 41

Completeness, and least upper

    bounds, 37–42

Composition of functions, 59

Connectedness, 150–52

Continuity, 57–80

    and compactness, 64–70

    definition, 57

    discontinuities, types of, 70–76

    epsilon-delta approach, 57–58

    global vs. local concept of, 59, 67

    semicontinuous functions, 152–58

    uniform, 66–70

Continuous functions, 57 ff.

Contradiction, proof by, 16–17

Contrapositive, 17

Convergence, 10–15

    pointwise, 117–20

    of power series, 52–55

    radius of, 54

    See also Pointwise convergence; Uniform convergence

Convergent subsequence, 33–34

Countable sets, 7–10

Countably infinite sets, 7–10

D

Decreasing function, 74, 85

Decreasing sequence, 41

Deleted open ball, 21

De Morgan laws, 5, 6, 27

Dense set, 68–69, 145–46

Derivative, 81 ff.

Diagonal process, 8, 46, 131, 163

Differentiable function, 81 ff.

Differentiation, 81–92

    definition, 81–83

    Generalized Mean Value Theorem, 85–86

    Mean Value Theorem, 83–85

    applications of, 86–91

Dini’s Theorem, 124–125

Direct image, 62–63

Disconnected sets, 150–51

Discontinuities, types of, 70–76

    of the first kind, 71–72

    infinite, 71

    jump, 71

    nonsimple, 73

    point, 71

    removable, 71

    of the second kind, 73–74

    simple, 71

Disjoint sets, 5–6, 151

Distance, 10–15

    from a point to a set, 79

Distance function, 10

Distributive law for sets, 2

Domain, 57

E

Empty set, 5

Epsilon-delta characterization, 57–58, 71, 73, 76

Equicontinuity, and Arzela-Ascoli Theorem, 130–33

Equicontinuous family of functions, 131

Euclidean metric, 10, 52

Euclidean plane, 2

Euclidean p-space, 11

Eventually (ev), 49

Everywhere continuous, nowhere differentiable function, 126–29

Existential quantifier, 18

Extended real numbers, 35

Extension:

    of a bounced continuous function, 143

    Tietze Extension Theorem, 143

    of a uniformly continuous function, 69

F

Finite sets, 7

Fσ set, 148

Function, 33

    bounded, 65, 125

    of bounded variation, 106–111

    continuous, 57 ff.

    decreasing, 74, 85

    differentiable, 81 ff.

    everywhere continuous and

    nowhere differentiable, 126–29

    increasing, 74, 85

    left-continuous, 111

    lower semicontinuous, 152

    monotone, 74

    right-continuous, 110

    semicontinuous, 152–58

    uniformly continuous, 66–70

    upper semicontinuous, 152

    variation of a, 106

Fundamental Theorem of Calculus, 104–5

    intuitive view of, 105

G

Generalized Mean Value Theorem, 85–86

Global concept of continuity, 59, 67

Greatest lower bound, 38

Gg set, 148

H

Heine-Borel Theorem, 31–32, 35, 130, 164

Homeomorphism, 144, 158

Horizontal line test for uniform convergence, 119–20

I

Image, 62–63

“Implies” connective, 15–16

Improper integrals, 114

Increasing function, 74, 85

Increasing sequence, 41

Induction, 19

Inductive procedure, 19

Infimum (inf), 38

Infinite discontinuity, 71

Infinitely often (i.o.), 49

Infinite sets, 7–9

Infs, properties of, 39–40

Integral (see Riemann-Stieltjes integral)

Integral test, 115

Integration by parts, 111–12

Interchange of operations, 117–19, 122–24

Interior of a set, 145

Intermediate Value Theorem, 75–76

    for derivatives, 87–88

Intersection, 2, 3, 6

Invalid interchange of operations, examples of, 117–19

Inverse image, 59

Irrational numbers, 147

Isolated point, 21

J

Jump discontinuity, 71

Jump function, 102–3

L

Largest subsequential limit, 46

Least upper bounds, 38

    and completeness, 37–42

Left-continuous function, 111

L’Hospital’s rule, 81, 88–90

lim inf, 46

Limit,

    definition of, 11

    lower, 46

    upper, 46

Limit concept, generalization of, 45–48

Limit operations, and uniform convergence, 122–25

Limit point, 21

lim sup, 46

Line integrals, and

    Riemann-Stieltjes integral, 104

Local concept of continuity, 59, 67

Local maximum, 83

Local minimum, 83

Logic, 15–21

    mathematical induction, 18–19

    negations, 19–21

    proof, types of, 16–17

    quantifiers, 17–18

    truth tables, 15–16

Lower bound, 38

Lower limit, 46

    properties of, 49

Lower semicontinuous (LSC) function, 152–58

Lower sum, 94

M

Mapping, 57

Mathematical induction, 18–19

Maximum, 65, 83

Mean Value Theorem, 81, 83–85

    applications of, 86–91

    Generalized, 85–86

    for integrals, 112–13

Metric, 10

Metric space, 10–13

    compactness criteria in, 162–64

Minimum, 65, 83

Monotone function, 74

Monotone sequence, 41, 124–25

Mutually exclusive sets, 5

N

Negations, 19–21

Neighborhood, 155

Nested Set Property, 29, 164–65

Nonsimple discontinuity, 73

“Not” connective, 15

Nowhere dense sets, 78, 144–45, 148

O

Open ball, 11

Open sets, 12

    continuity and, 61–62

    relation between closed sets and, 13–14

    unions and intersections of, 25–28

Open subsets of R, 42

“Or” connective, 15

Order of summation, reversal of, 137–39

P

Partition, 93

    refinement of, 95

    size of, 93

Path-connectedness, 150–51

Perfect set, 78

Piece wise continuous function, 99

Point discontinuity, 71

Pointwise bounded sequence of functions, 132

Pointwise convergence, 117–20

    vertical line test for, 119–20

Power series, convergence of, 52–55

Predicate, 17–18

Preimage, 59–61

Probability, and Riemann-Stieltjes integral, 104

Proof:

    by cases, 22

    by contradiction, 17

    via contrapositive, 17

    types of, 17

Proper subset, 4

Proposition, 15

Q

Quantifiers, 17–18, 19

R

Radius of convergence, 54

Ratio test, 53

Rational numbers, 7, 147

Real numbers, 1, 165

    extended, 35

    upper/lower limits of sequences of, 45–56

Refinement, 95

Relative topology, 142

Removable discontinuity, 71

Riemann integral, 93 ff.

Riemann-Stieltjes integral, 93–116

    change of variable formula, 112

    definitions, 93–96

    evaluation formula for, 103

    existence of, 96

    improper, 114

    integration by parts, 111–12

    line integrals and, 104

    Mean Value Theorem for, 112–13

    probability and, 104

    properties of, 98–106

    upper bounds on, 113

Riemann-Stieltjes sum, 94

Right-continuous function, 110

Rolle’s Theorem, 83–86

Root test, 53–55

S

Semicontinuous functions, 152–58

Separated sets, 151

Sequence, 33–34, 45–52

    bounded, 34

    Cauchy, 24, 40–41, 124, 164

    convergence of, 10–15

    and limit concept, 45

    monotone, 41, 124–25

    unbounded, 36

Sequences of real numbers, upper and lower limits of, 45–56

Sets, 1 ff.

    bounded, 29–35, 161–62

    Cantor, 76–80

    of category 1 and category 2, 145

    closed, 12–14, 25–28

    closure of, 21–22

    co-finite, 23

    compact, 30

    complement of, 2, 3, 6

    connected, 150–52

    countable, 7–10

    dense, 68, 69, 145–46

    disjoint, 5–6, 151

    distributive law for, 2

    empty, 5

    finite, 7

    Fσ, 148

    Gg, 148

    infinite, 7

    intersection of, 2, 3, 6

    mutually exclusive, 5

    nested set property, 29, 164–65

    nowhere dense, 78, 144–45, 148

    open, 11, 25, 26, 42

    path-connected, 150–51

    perfect, 78

    separated, 151

    thin, 144

    totally bounded, 161–62

    totally disconnected, 78

    uncountable, 7

    uncountably infinite, 7

    union of, 2

Set-theoretic difference, 6

Simple discontinuity, 71

Smallest subsequential limit, 46

Squeeze theorem, 15

Stone-Weierstrass Theorem, 137

Strong induction, 19

Subcovering, 30

Subsequence, 33–34

    diagonal, 131

Subsequential limit, 46

Subset, 4, 8

    proper, 4

Supremum (sup), 38

Sups, properties of, 39–40

T

Taylor’s formula with remainder, 81,90–92

Thin sets, 144

Tietze Extension Theorem, 143

Topological properties, 25 ff., 141 ff.

Topologist’s sine curve, 151

Totally bounded set, 161–62

Totally disconnected set, 78

Triangle inequality, 10

True by default, 16

Truth tables, 15–16

U

Unbounded sequence, 36

Uncountable set, 7

Uncountably infinite set, 8–10

Uniform continuity, 66–70

Uniform convergence, 117–40

    definition, 119–20

    horizontal line test for, 119–20

    and limit operations, 122–25

    of polynomials, 134–37

    of power series, 129

    of series, 125–29

Uniformly bounded sequence of functions, 125, 132

Uniformly continuous function, 66–70

Union of sets, 2, 3, 6

Uniqueness argument, 50

Universal quantifier, 18

Universe, 2

Upper bound, 38

Upper bounds on integrals, 113

Upper limit, 46

    properties of, 49

Upper semicontinuous (USC), function, 152–58

Upper sum, 94

Urysohn’s Lemma, 141

V

Vacuously true, 16

Variation of a function, 106

Venn diagrams, 2–5

Vertical line test, 119–20

w

Weierstrass Approximation Theorem, 134–37

Weierstrass M-test, 126

Well-ordering principle, 19