Chapter 4

Question 1

continuity of a function from a metric space (X, d_X) to a metric space (Y, d_Y)

Answer 1

a function f: R→R is _______ at x=a if

1. the function is defined at a
2. the limit of f(x) as x approaches a exists and
3. the limit of f(x) as x approaches a is equal to f(a)
4. If a is not a limit point of X, then f is continuous at a

Question 2

limit of a function from a metric space (X, d_{<span>X</span>}) to a metric space (Y, d_Y)

Answer 2

Let (X,dX) and (Y, dY) be metric spaces and f: K→Y where K is contained in X. Let a be a limit point of the domain K. We say that the limit of f(x) as x→a= L if for all epsilon>0, there exists a delta>0 s.t . if 0<d>y(f(x),L)<epsilon>
</epsilon></d>

Question 3

Let X be a metric space and let S ⊆ X. Then S is closed if and only if

Answer 3

X\S is open.

Question 4

Every intersection of closed sets is ____, and every _______ ________of closed sets is ________

Answer 4

closed; finite union; closed.

Question 5

closure of S cl(S)

Answer 5

Let S be a subset of a metric space X. The intersection of all closed subsets of X that contain S

Question 6

Theorem Let S be a subset of a metric space X. Then

Answer 6

1. S is a closed set. 2. S is closed if and only if S = S. 3. S = S ∪ {x ∈ X : x is a limit point of S}

Question 7

Theorem Let x ∈ X, where X is a metric space, and let S ⊆ X. TFAE:

Answer 7

1. x ∈ S
2. There exists a sequence in S that converges to x. (How is this different than x being a limit point?? Be sure to understand the distinction).
3. Every open ball about x contains a point of S.
4. Every open set containing x contains a point of S.

Question 8

Let X be a metric space and S ⊆ X. A point x ∈ S is called an _____ _____ of S if there is an open ball about x that is contained in S.

Answer 8

interior point of S

Question 9

The set of all _______ ________of S is called the ______ of S and is denoted by int(S).

Answer 9

interior points; interior

Question 10

Theorem about interior of S. Let X be a metric space and let S ⊆ X. Then,

Answer 10

1. int(S) is an open set 2. Every open set of X that is contained in S is contained in int(S). 3. The union of all open subsets of X that are contained in S is equal to int(S). 4. S is open if and only if S = int(S) 5. int(S) = X\ cl(X\S) (whoa…)

Question 11

boundary of S

Answer 11

Let X be a metric space and S ⊆ X. We define the ________ to be S ∩ cl(X\S), and we denote it by ∂(S).

Question 12

Theorem about boundaries. Let X be a metric space and S ⊆ X. Then,

Answer 12

1. ∂(S) is a closed set 2. An element x ∈ X is in ∂(S) if and only if for every r > 0, Br(x) ∩ S 6= ∅ and Br(x) ∩ (X\S) 6= ∅. 3. S is closed if and only if S contains ∂(S). 4. S is open if and only if S ∩ ∂(S) = ∅

Question 13

Theorem 4.3.3 usual set-up TFAE:

Answer 13

1. f is continuous at a
2. For all epsilon>0, there exists a delta s.t. if 0<d>x(x,a)<delta>y(f(x),f(a))<epsilon>
   </epsilon></delta></d><li>if (a_n) is a sequence in X and converges to a set of distinct points, then f(a_n)→f(a)</li>

</epsilon></delta></d>

Question 14

Theorem 4.3.4

Answer 14

Let X, Y, Z be metric spaces. If f: X→Y and g:Y→Z and both functions are continuous, then the gºf: X→Z is also continuous

Question 15

Theorem 4.3.5 Think about continuity

Answer 15

Let X & Y be metric spaces and f: X→Y. f is continuous ⇔ TFSIT: if U is open in Y, the the inverse image of [U] is open in X

Question 16

Theorem 4.2.3 Uniqueness Theorem

Answer 16

Let X and Y be metric spaces and let K be contained in X and f: K→Y. Let a be a limit point of K. Then, the limit of f(x) as x approaches a is unique

Question 17

Theorem 4.2.4 Limit Points

Answer 17

Let X and Y be metric spaces. Let f: X→Y and a be a limit point of X. Let L be an element of Y. TFAE:

1. the limit of f(x) as x approaches a is L
2. for all sequences (a_n) in X of distinct points, if a_n→a then f(a_n)→L