Continuity and Connectedness

Question 1

Continuous Function (at x0)

Answer 1

• Let f:X->Y be continuous at x_0 in X.
• Thus for all e > 0, there exists a delta > 0 such that d_Y(f(x),f(x_0)) < e whenever d_X(x,x_0) < delta.
• In other words, f(B_X(x_0, delta)) is contained in B_Y(f(x_0), e)

Question 2

Continuous Function (at x0) Equivalences

Answer 2

• Let f:X->Y, x0 in X. The following are equivalent:
  1. f is continuous at x_0.
  2. Whenever (x_n)(n=1,infinity) in X with x_n _. x_0 as n-> infinity, f(x_n) -> f(x_0) as n -> infinity
  3. For any open set V in (Y, d_Y) with f(x_0) in V, there exists an open set U in (X, d_X) with x_0 in U and f(U) contained in V.

Question 3

Continuous Function (Everywhere) Equivalences

Answer 3

• Let f:X->Y. The following are equivalent:
  1. f is continuous.
  2. f^-1(V) is open in (X, d_X) for any open set V contained in (Y, d_Y).
  3. f^-1(E) is closed in (X, d_X) for any closed set V contained in (Y, d_Y).

Question 4

Uniformly Continuous Function

Answer 4

• Let f:X->Y be a function.
• f is uniformly continuous if for all e > 0, there exists a delta > 0: d_Y(f(x),f(x’)) < e whenever d_X(x,x’) < delta.
• Essentially, can bound the distance between f(x) and f(x’), and x and x’ by some epsilon.

Question 5

Uniformly Continuity and Continuity in Compact Space

Answer 5

• Let (X, d_X), (Y, d_Y) be metric spaces where (X, d_X) is compact.
• Then f:X->Y is uniformly continuous iff f is continuous.

Question 6

Disconnected

Answer 6

• A metric space (X, d) is disconnected if there exists disjoint, nonempty open sets V and W in (X,d) such that X = VuW.
• Thus:
  1. V, W are open.
  2. V intersect W is empty.
  3. X = VuW

Question 7

Connected

Answer 7

• A space that is not disconnected.

Question 8

Connectedness and Intervals in R

Answer 8

• Let E contained in R be a metric space. TFAE:
  1. E is connected.
  2. E is an interval (meaning for a, b in R, a =/ b: E = [a,b] or [a,b) or (a,b] or (a,b).

Question 9

Connectedness and Clopen Sets

Answer 9

• Let (X,d) be a metric space. TFAE:
  1. (X,d) is connected.
  2. The empty set and X are the only sets in (X,d) which are clopen.