Real Variable Theory

Question 1

Axiom of Completeness

Answer 1

Every nonempty set of real numbers that is bounded above has a least upper bound aka supremum

Question 2

Metric space

Answer 2

Let X be a set. We say that the pair (X,d) is a metric space with metric d if 1. d(x,y) = d(y,x) 2. d(x,y) = 0 iff x = y 3. d(x,z) < d(x,y) + d(y,z)

Question 3

Lemma (u- epsilon < a)

Answer 3

Let A be a nonempty subset of |R. Then a real number u∈|R is the sup(A) iff 1. u is an upper bound for A and 2. for each epsilon > 0, there exists a member a∈A s.t. u- epsilon < a

Question 4

Absolute value/basic metric space

Answer 4

d(x,y) = |x - y|

Question 5

Bounded metric space

Answer 5

A ⊆X in a metric space (X,d) is called bounded if there exists a point x∈X and a pos. real number r such that A ⊆ {p in X | d(p,x) < r} = B_r(x) (ball of radius r centered at x)

Question 6

Archimedean Property of |R

Answer 6

Let x be any real number. 1. Then there exists an n∈|N such that n > x and 2. for any y > 0, there exists n∈|N such that 1/n < y
(Proof of 1. BWOC, n

Question 7

Taxicab metric

Answer 7

d(x,y) = |x1 - y1| + |x2 - y2|

Question 8

Discrete metric

Answer 8

d(x,y) = {0 if x=y; 1 if x neq y}

Proof of triangle: 2 cases, x = z and x neq z

Question 9

Convergent Seq (in metric space)

Answer 9

Let (a_n) be a sequence in (X,d). We say that a_n –> p in X if for all epsilon > 0 there exists N in |N such that d(a_n, p) < epsilon whenever n > N

Question 10

Limit of a Convergent Seq is Unique (Prop)

Answer 10

Let (X,d) be a metric space and let a_n be a convergent seq in X. Then the limit of (a_n) is unique if a_n -> p and a_n -> q, then p = q.
(Proof: BWOC p neq q, use triangle inequality and let epsilon/2)

Question 11

Monotone Convergence Theorem

Answer 11

Let (a_n) be a sequence in (|R , |.|) such that 1. a_n is bounded above and 2. a_n is increasing. Then a_n is convergent. 
(Proof: Let A = {a_n}, use givens and property of sup so there exists an a_m in A s.t. 
s - ep < a_m. WTS s - ep < a_n, let m = N.)

Question 12

Subsequence

Answer 12

Let a_n be any sequence. We say b_n is a subsequence of a_n if b_n = a_f(n) where f is an increasing function f: |N –> |N

Question 13

Proposition of all subsequences of a convergent seq converge to same limit

Answer 13

If a_n is a convergent sequence, then any subsequence of a_n is convergent to the same limit as a_n
(Proof: suppose a_n -> p. WTS b_n –> p. use def of converg in met space, works cuz f is increasing f(n) > n)

Question 14

Bolzano-Weierstrass

Answer 14

Let a_n be any bounded sequence in |R. Then a_n has a convergent subsequence

Question 15

Nested Interval Property

Answer 15

Consider the closed (bounded) intervals
I_n = [a_n, b_n] for a_n, b_n in |R. Suppose that these intervals are nested in the sense that I1 >(ss) I2 > I3 >… Then the intersection of I_n is not empty.
(Proof: consider a_n of left endpoints of intervals; a_n is increasing and bounded, MCT; converges to a = sup…)

Question 16

Open Set

Answer 16

A ⊆ X is open in X if for all a in A there exists an r > 0 such that B_r(a) ⊆ A

Question 17

Closed Set

Answer 17

A subset X is closed if A^c = X - A is open in X A subset A is a closed subset of X if it contains all its limit points (A complement aka everything but A)

Question 18

Limit Point

Answer 18

A point x in X is a limit point for A ⊆ X if there is a sequence a_n in A s.t. a_n→x and a_n neq x for all n in |N

Question 19

Closure

Answer 19

Closure of A, denoted by A(bar) = A U A’

Question 20

Isolated Point

Answer 20

a in A is an isolated point if a is not a limit point

Question 21

Following are open in (X,d)…

Answer 21

X (metric space X), nonempty set, (0,1) in |R, 1st quadrant in |R^2

Question 22

Following are closed in (X,d)…

Answer 22

X (b/c X^c = nonempty = open), nonempty set, [0,1] in |R

Question 23

Sequentially Compact

Answer 23

Let (X,d) be a metric space. We say that A subset X is sequentially compact if any sequence in A has a convergent subsequence that converges to a point that is also in A

Question 24

Heine-Borel Theorem

Answer 24

Consider (|R, |.|). Then A ⊆ |R is sequentially compact iff A is closed and bounded

Question 25

Theorem, Any compact set is closed and bounded

Answer 25

Let (X,d) be a metric space and let A be a sequentially compact subset of X. Then A is bounded and closed in X.
(Boundedness proof: Assume A is seq compact but not bounded; use inverse of bounded metric space; convergent subseq.
Closeness proof: Assume A is seq compact but not closed; A^c is not open; use def of open; then find a_n in A with 1/n; convergent subseq)

Question 26

Compact

Answer 26

A set K ⊆ R is compact if every sequence

in K has a subsequence that converges to a limit that is also in K.

Question 27

An important property of sup

Answer 27

For each n>0 there exists a sequence a_n in A such that s - 1/n

Question 28

Disconnected Set

Answer 28

Let (X,d) be a metric space and let A⊆X. Suppose there exists open sets U and V s.t.
(1) U n V = empty set (2) U n A neq empty set and V n A neq empty set (3) A⊆U u V

Question 29

Connected Set

Answer 29

If A⊆X is not disconnected, then it is called connected

Question 30

Theorem (on Compact)

Answer 30

A subset A of a metric space is compact if and only if it is sequentially compact

Question 31

Theorem (also on compact)

Answer 31

Suppose A⊆X. If A is compact, then A is closed and bounded

Question 32

Interior point

Answer 32

Suppose A⊆X. A point a in A is an interior point of A if there is an r>0 s.t. B_r(a) ⊆ A. The set of all interior points is denoted by A^o

Question 33

Dirichlet Function

Answer 33

f(x) = {1 if x in Q; 0 if x not in Q

Question 34

epsilon − δ Functional Limit

Answer 34

Let A be a subset of (|R, |.|), and let a be a limit point of A. Suppose f: A→|R. We say
lim x→a f(x) = L for some L in |R if for all epsilon >0 there exists some δ>0 such that |f(x) - L| < epsilon whenever |x-a|< δ

Question 35

Continuous function

Answer 35

Let A⊆|R and let f: A → |R be a function. We say that f is continuous at x=a if for all epsilon > 0 there exists δ>0 such that |f(x) - f(a)| < epsilon whenever |x-a| < δ.

Question 36

Proposition (continuity/limit point)

Answer 36

Let f: A→|R and let a be a limit point of A. Then the following are equivalent: (i) f is continuous at x=a (ii) lim x→a f(x) = f(a)
(Proof: => use def of continuity; limit point
<=use given to get def of continuity)

Question 37

Proposition (continuity/sequence/limit)

Answer 37

Let f: A→|R and let a in A. TFAE: (i) f is continuous at x=a (ii) For each sequence (a_n) in A such that a_n → a we have 
lim f(a_n) = f(a)

Question 38

Proposition (continuity/sequence)

Answer 38

Let f: A→|R and let a in A. TFAE: (i) f is continuous at x=a (ii) For each sequence (a_n) in A such that a_n → a we have
f(a_n) → f(a)

Question 39

Algebraic Continuity Theorem

Answer 39

Assume f :A→R and g : A → R are continuous at point c ∈ A. Then,

(i) kf(x) is continuous at c for all k ∈ R;
(ii) f(x) + g(x) is continuous at c;
(iii) f(x)g(x) is continuous at c; and
(iv) f(x)/g(x) is continuous at c, provided the quotient is defined.

Question 40

Algebraic Limit Theorem for Functional Limits

Answer 40

Let f and g be functions defined on a domain A ⊆ R, and assume limx→c f(x) = L
and limx→c g(x) = M for some limit point c of A. Then, (i) limx→c kf(x) = kL for all k ∈ R,
(ii) limx→c [f(x) + g(x)] = L + M,
(iii) limx→c [f(x)g(x)] = LM, and
(iv) limx→c f(x)/g(x) = L/M, provided M = 0.

Question 41

Grand Theorem (dont need to know name)

Answer 41

Let A⊆|R and let f: A→|R be a function. Then f is continuous on A iff for any open set U⊆|R in the domain, f^-1(U) is open in the domain.

Question 42

Corollary (compact)

Answer 42

f(compact) = compact implies that f: [0,1]→|R then f([0,1]) is compact in |R, f([0,1]) has a max/min

Question 43

Theorem (closed set)

Answer 43

f: A→|R is continuous iff f^-1(K) is closed for any closed set K in |R.

Question 44

Theorem (continuous image of compact set)

Answer 44

If K⊆|R is compact and f: K→|R is continuous, then f(K) is compact

Question 45

Extreme Value Theorem

Answer 45

Suppose f: |R→|R is continuous and let K be a compact subset of |R. Then f:K-|R attains its max and min on K. In other words, there exists x_0, x_1 in K such that f(x_0)

Question 46

Notes on [0,1]..

Answer 46

f is continuous on [0,1] and [0,1] is compact

Question 47

Theorem (continuous image)

Answer 47

“The continuous of image of a connected set is connected”

Let f:|R→|R be continuous and suppose C is a connected subset of |R. Then f(C) is connected

Question 48

Intermediate Value Theorem

Answer 48

Let a and b be real numbers and a

Question 49

Fixed Point

Answer 49

A point x ∈ X is called a fixed point of a function f : X → X if f(x) = x

Question 50

Uniformly continuous on A

Answer 50

f: A→|R is uniformly continuous on A is for all epsilon > 0, for all x,a in A there exists a
delta > 0 such that |f(x) - f(a)| < epsilon whenever |x-a| < delta

Question 51

Criterion for lack of Uniform Continuity

Answer 51

f: A→|R is NOT uniformly continuous on A iff there exists sequences (x_n) and (a_n) in A such that (x_n - a_n) → 0 but there exists an epsilon > 0 for which | f(x_n) - f(a_n) | > epsilon for sufficiently large n

Question 52

Two observations of uniform contin.

Answer 52

(1) Uniform continuity only implies continuity (not the other way)
(2) If f is uniformly continuous on A, then f is uniformly continuous on any subset B⊆A

Question 53

Theorem (uniform contin/compact)

Answer 53

Let K be compact and f: K→|R continuous. Then f is uniformly continuous on K

Question 54

Thomae’s Function

Answer 54

t(x) = { 1 if x=0; 1/n if x = m/n in Q;

0 if x not in Q}

Question 55

Fact on continuity

Answer 55

Any polynomial is continuous

Question 56

Limit definition of derivative

Answer 56

Let A be any interval. We say f: A →|R is differentiable at a in A if
f’(a) = limx→a f(x) − f(a) / x − a

Question 57

Proposition (differentiable/continuous)

Answer 57

If f: A→|R is differentiable at x=a, then f is indeed continuous at x=a

Question 58

Interior Extremum Value Theorem

Answer 58

If f is differentiable and it has a local max (or a local min) at x = a, then f’(a) = 0

Question 59

Rolle’s Theorem

Answer 59

If f: [a,b] →|R is continuous on [a,b] and f is differentiable on (a,b) and f(a) = f(b), then
f’(c) = 0 for some c in (a,b)

Question 60

Mean Value Theorem

Answer 60

Suppose f: [a,b] is continuous on [a,b] and f is differentiable on (a,b). Then there exists some c in (a,b) such that f’(c) = f(b) − f(a) / b − a

Question 61

Consequence of MVT

Answer 61

If f: (a,b)→|R is a function such that f’(x) = 0 for all x in (a,b). Then f(x) = c (constant)

Question 62

Product Rule for Differentiation

Answer 62

Suppose f and g are differentiable at some point a in A. Then f.g defined by f.g(x) = f(x)g(x) is also differentiable at x = a and moreover (f.g)’(a) = f’(a)g(a) + f(a)g’(a)

Question 63

Cauchy Completeness

Answer 63

Let (X,d) be a metric space. We say that (X,d) is Cauchy Complete if any Cauchy sequence in X is convergent in X

Question 64

Cauchy Sequence

Answer 64

Let (X,d) be a metric space and let (x_k) be a sequence in X. We say (x_k) is a Cauchy sequence if for all epsilon > 0 there exists N in |N such that whenever m and n satisfy m>N and n>N we have d(x_m, x_n) < epsilon

Question 65

Big picture (Cauchy)

Answer 65

(1) Any convergent sequence is a Cauchy sequence in any metric space (2) Any Cauchy sequence is a convergent sequence in Cauchy complete spaces (3) (|R, |.|) is Cauchy complete so in |R Cauchy seq = convergent seq

Question 66

Important Lemma (Cauchy/convergent)

Answer 66

Let (X,d) be a metric space and let (x_k) be a Cauchy sequence in X. Suppose (x_k) has a convergent subsequence (x_ki) and that x_ki→x in X. Then x_k is also convergent to x.

Question 67

Theorem

Answer 67

(|R, |.|) is Cauchy complete

Question 68

Theorem (Cauchy seq/bounded)

Answer 68

Let (X,d) be a metric space and let (a_n) be a Cauchy sequence in X. Then {a_n | n in |N} is a bounded subset of X
(Proof: def of Cauchy seq; let ep = 1 and n = N+1; use A = BUC, bounded}

Question 69

Partition

Answer 69

A partition P for the interval [a,b] is a finite set, P = {x_0,…,x_n} such that
x_0 = a < x1 < x2

Question 70

Upper/Lower Sum

Answer 70

Let f be a (bounded) function on [a,b] and let P be a partition of [a,b]. We define upper/lower sums of f with respect to P as follows, P = {x_0,…,x_n}:
U(f,P) = sumM_k(x_k-x_k-1) where M_k = {f(x)|x in [x_k-1, x_k]}
L(f,P) = summ_k(x_k - x_k-1) where m_k = {f(x)|x in [x_k-1, x_k]}

Question 71

Proposition (upper/lower sum)

Answer 71

For any partition P, L(f,P)

Question 72

Refinement

Answer 72

Let f be bounded on [a,b] and let P and Q be partitions of f on [a,b]. We say Q is a refinement of P if P is a subset of Q

Question 73

Integrable

Answer 73

Let f be bounded on [a,b]. We say that f is integrable on [a,b] if U(f) = L(f) where U(f) = inf{U(f,P) | P partition} and L(f) = sup{L(f,P) | P partition}

Question 74

Lemma (2 paritions)

Answer 74

If P1 and P2 are any two partitions of [a,b], then L(f, P1)

Question 75

Prop (refinement/parition)

Answer 75

If Q is a refinement of P, then L(f,P) eq U(f,Q)

Question 76

Darboux’s Criterion for Riemann Integrability

Answer 76

Let f be a bounded function on [a,b]. Then f is integrable on [a,b] iff for all epsilon > 0, there exists some partition P_ep such that U(f,P_ep) - L(f,P_ep) < epsilon

Question 77

Theorem (contin/integrable)

Answer 77

Any continuous function f on [a,b] is integrable

Question 78

Fundamental Theorem of Calculus

Answer 78

Let f be a differentiable function. Assume f’ is integrable on [a,b]. Then integral from a to b f’(x)dx = f(b) - f(a)