Chapter 4 Flashcards
continuity of a function from a metric space (X, dX) to a metric space (Y, dY)
a function f: R→R is _______ at x=a if
- the function is defined at a
- the limit of f(x) as x approaches a exists and
- the limit of f(x) as x approaches a is equal to f(a)
- If a is not a limit point of X, then f is continuous at a
limit of a function from a metric space (X, d<span>X</span>) to a metric space (Y, dY)
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>
Let X be a metric space and let S ⊆ X. Then S is closed if and only if
X\S is open.
Every intersection of closed sets is ____, and every _______ ________of closed sets is ________
closed; finite union; closed.
closure of S cl(S)
Let S be a subset of a metric space X. The intersection of all closed subsets of X that contain S
Theorem Let S be a subset of a metric space X. Then
- 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}
Theorem Let x ∈ X, where X is a metric space, and let S ⊆ X. TFAE:
- x ∈ S
- 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).
- Every open ball about x contains a point of S.
- Every open set containing x contains a point of S.
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.
interior point of S
The set of all _______ ________of S is called the ______ of S and is denoted by int(S).
interior points; interior
Theorem about interior of S. Let X be a metric space and let S ⊆ X. Then,
- 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…)
boundary of S
Let X be a metric space and S ⊆ X. We define the ________ to be S ∩ cl(X\S), and we denote it by ∂(S).
Theorem about boundaries. Let X be a metric space and S ⊆ X. Then,
- ∂(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) = ∅
Theorem 4.3.3 usual set-up TFAE:
- f is continuous at a
- 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>
Theorem 4.3.4
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
Theorem 4.3.5 Think about continuity
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