Test 2

Question 1

Cantor Set

Answer 1

a set Cn consisting of 2^n closed intervals each having length 1/(3^&n). Finally, we define the Cantor set C (Fig. 3.1) to be the intersection of all the Cns

Properties:
C is uncountable
C contains only the endpoints of each sub interval
C is a perfect set

Question 2

open set

Answer 2

A set O ⊆ R is open if for all points a ∈ O there exists an ε-neighborhood Vε(a) ⊆ O

Question 3

limit point

Answer 3

A point x is a limit point of a set A if every ε-neighborhood Vε(x) of x intersects the set A at some point other than x.

x is quite literally the limit of a sequence in A

Question 4

isolated point

Answer 4

A point a ∈ A is an isolated point of A if it is not a limit point of A

Question 5

closed set

Answer 5

A set F ⊆ R is closed if it contains its limit points

Question 6

closure

Answer 6

Given a set A ⊆ R, let L be the set of all limit points of A. The closure of A is defined to be A-closure = A ∪ L.

Question 7

complement

Answer 7

Ac = {x ∈ R : x /∈ A}.

A set O is open if and only if Oc is closed. Likewise, a set F is closed if and only if Fc is open

Question 8

compact set

Answer 8

A set K ⊆ R is compact if and only if it is closed and bounded.

Question 9

open cover

Answer 9

An open cover for A is a (possibly infinite) collection of open sets {Oλ : λ ∈ Λ} whose union contains the set A; that is, A ⊆ UOλ.

Given an open cover for A, a finite subcover is a finite subcollection of open sets from the original open cover whose union still manages to completely contain A.

A set is compact if every open cover of A contains a finite subcover of A

Question 10

perfect set

Answer 10

A set P ⊆ R is perfect if it is closed and contains no isolated points.

Closed intervals (other than the singleton sets [a, a]) serve as the most
obvious class of perfect sets, but there are more interesting examples.

Question 11

separated sets

Answer 11

Two nonempty sets A, B ⊆ R are separated if A-closure ∩ B and
A ∩ B-closure are both empty

Question 12

connected sets

Answer 12

A set that is not disconnected is called a connected set.

Question 13

disconnected set

Answer 13

A set E ⊆ R is disconnected if it can be written as E = A ∪ B, where A and B are nonempty separated sets.

Question 14

Fσ sets

Answer 14

A set A ⊆ R is called an Fσ set if it can be written as the countable union of closed sets

Question 15

Gd set

Answer 15

A set B ⊆ R is called a Gd set if it can be
written as the countable intersection of open sets.

Question 16

nowhere-dense sets

Answer 16

A set E is nowhere-dense if E-closure contains no nonempty open intervals.

Question 17

limit of a function

Answer 17

Let f : A → R, and let c be a limit
point of the domain A. We say that limx→c f(x) = L provided that, for all
ε > 0, there exists a δ > 0 such that whenever 0 < |x − c| < δ (and x ∈ A) it
follows that |f(x) − L| < ε

Question 18

continuous

Answer 18

A function f : A → R is continuous at a
point c ∈ A if, for all ε > 0, there exists a
δ > 0 such that whenever |x − c| < δ
(and x ∈ A) it follows that
|f(x) − f(c)| < ε

If f is continuous at every point in the domain A, then we say that f is continuous on A

Question 19

uniform continuity

Answer 19

A function f : A → R is uniformly
continuous on A if for every ε > 0 there exists a δ > 0 such that for all x, y ∈ A,
|x − y| < δ implies |f(x) − f(y)| < ε

Question 20

intermediate value property

Answer 20

A function f has the intermediate value property on an interval [a, b] if for all
x < y in [a, b] and all L between f(x) and f(y), it is always possible to find a point c ∈ (x, y) where f(c) = L

Question 21

sets of discontinuity (Df)

Answer 21

Given a function f : R → R, define Df ⊆ R to be the set of points where the function f fails to be continuous

Question 22

monotone, increasing, decreasing functions

Answer 22

A function f : A → R is increasing on A if f(x) ≤ f(y) whenever x < y

decreasing if f(x) ≥ f(y) whenever x < y

A monotone function is one that is either increasing or decreasing.

Question 23

removable discontinuity

Answer 23

If limx→c f(x) exists but has a value different from f(c), the discontinuity
at c is called removable

Question 24

jump discontinuity

Answer 24

If limx→c+ f(x) =/= limx→c− f(x),
then f has a jump discontinuity at c

Question 25

essential discontinuity

Answer 25

If limx→c f(x) does not exist for some other reason, then the discontinuity
at c is called an essential discontinuity.

Question 26

α-continuous

Answer 26

Let f be defined on R, and let α > 0. The function f is α-continuous at x ∈ R if there exists a δ > 0 such that for all y, z ∈ (x−δ, x+δ) it follows that |f(y) − f(z)| < α

Question 27

Df-α

Answer 27

Df-alpha = {x ∈ R : f is not α-continuous at x}.

Question 28

Characterization of Compactness in R

Answer 28

A set K ⊆ R is compact if and only if it is closed and bounded

Question 29

Nested Compact Set Property

Answer 29

If K1 ⊇ K2 ⊇ K3 ⊇ K4 ⊇ …
is a nested sequence of nonempty compact sets, then the infinite intersection Kn is not empty.

Question 30

A nonempty perfect set is

Answer 30

uncountable

Question 31

Sequential Criterion for Functional Limits

Answer 31

Given a function f : A → R and a limit point c of A, the following two statements are
equivalent:
(i) lim (x→c) f(x) = L

(ii) For all sequences (xn) ⊆ A satisfying xn =/= c and (xn) → c, it follows that f(xn) → L.

Question 32

Characterization of Continuity

Answer 32

The function f is continuous at c if and only if any one of the following
three conditions is met:

(i) For all ε > 0, there exists a δ > 0 such that |x−c| < δ (and x ∈ A) implies
|f(x) − f(c)| < ε

(ii) For all V(f(c)), there exists a Vδ(c) with the property that x ∈ Vδ(c) (and x ∈ A) implies f(x) ∈ Vε(f(c))

(iii) For all (xn) → c (with xn ∈ A), it follows that f(xn) → f(c)

If c is a limit point of A, then the above conditions are equivalent to
(iv) lim x→c f(x) = f(c).

Question 33

Algebraic Limit Theorem for Functional Limits

Answer 33

(i) lim x→c kf(x) = kL for all k ∈ R,
(ii) lim x→c [f(x) + g(x)] = L + M,
(iii) limx→c [f(x)g(x)] = LM
(iv) lim x→c f(x)/g(x) = L/M, M =/= 0

Question 34

Divergence criterion for functional Limits

Answer 34

Let f be a function defined on A, and let c be a limit point of A. If there exist two
sequences (xn) and (yn) in A with
xn =/= c and yn =/= c and
lim xn = lim yn = c
but lim f(xn) =/= lim f(yn),

then we can conclude that the functional limit lim x→c f(x) does not exist

Question 35

Criterion for Discontinuity

Answer 35

Let f : A → R, and let c ∈ A be a limit point of A. If there exists a sequence (xn) ⊆ A where (xn) → c but such that f(xn) does not converge to f(c), we may conclude that f is not continuous at c.

Question 36

Algebraic Continuity Theorem

Answer 36

Assume f : A → R and g : A → R are continuous at a 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 37

Composition of Continuous
Functions

Answer 37

Given f : A→R and g : B → R, assume that the range f(A) = {f(x) : x ∈ A} is contained in the domain B so that the composition g ◦ f(x) = g(f(x)) is defined on A. If f is continuous at c ∈ A, and if g is continuous at f(c) ∈ B, then g ◦ f is
continuous at c.

Question 38

Preservation of Compact Sets

Answer 38

Let f : A → R be continuous on A. If K ⊆ A is compact, then f(K) is compact as well.

Question 39

Extreme Value Theorem

Answer 39

If f : K → R is continuous on
a compact set K ⊆ R, then f attains a maximum and minimum value.

In other words, there exist x0, x1 ∈ K such that f(x0) ≤ f(x) ≤ f(x1) for all x ∈ K.

Question 40

Sequential Criterion for Absence of Uniform Continuity

Answer 40

A function f : A → R fails to be uniformly continuous on A if and only if there exists a particular ε0 > 0 and two sequences (xn) and (yn) in A satisfying

|xn − yn| → 0 but |f(xn) − f(yn)| ≥ ε0.

Question 41

A set containing only isolated points must be closed

Answer 41

False

Question 42

The set Z is a nowhere-dense set

Answer 42

True

Question 43

Let A be a set and c element of A be an isolated point. Then

lim x->c f(x) = f(a)

Answer 43

False

Question 44

The function f(x) = x^2 is uniformly continuous on [0, 1)

Answer 44

False

Question 45

Suppose A is an open set. Then A-closure is perfect

Answer 45

True

Question 46

Let f(x) : R -> R be a continuous function, and suppose for any x except possibly 1, f(x) > 0. Then, f(1) >= 0

Answer 46

True

Question 47

Let f : R ! R be a function. If x ∈ Df , then x ∈ Df-a for every alpha.

Answer 47

False

Question 48

Let (Ai) be a countable sequence of open sets. Then
The closure of the union of all Ai is equal to the union of all Ai closures

Answer 48

False

Question 49

If A is open and unbounded, then Ac is compact

Answer 49

False