Chapter 4 Flashcards

1
Q

continuity of a function from a metric space (X, dX) to a metric space (Y, dY)

A

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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

limit of a function from a metric space (X, d<span>X</span>) to a metric space (Y, dY)

A

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>

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

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

A

X\S is open.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

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

A

closed; finite union; closed.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

closure of S cl(S)

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

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

A
  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}
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

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

A
  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.
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

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.

A

interior point of S

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

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

A

interior points; interior

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

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

A
  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…)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

boundary of S

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

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

A
  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) = ∅
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Theorem 4.3.3 usual set-up TFAE:

A
  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 (an) is a sequence in X and converges to a set of distinct points, then f(an)→f(a)</li>

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Theorem 4.3.4

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Theorem 4.3.5 Think about continuity

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Theorem 4.2.3 Uniqueness Theorem

A

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

17
Q

Theorem 4.2.4 Limit Points

A

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 (an) in X of distinct points, if an→a then f(an)→L