Theorems

For some function continuous, the Extreme Value Theorem and Intermediate Value Theorem apply. More generally, for continuous, we get

  • is compact is compact.
  • is connected is connected.

Recall that the Heine-Borel Theorem gives us .

Last Time

Def: metric space:

seq. Say converges to .If such that .

Def:

We say is continuous at if sequence that converges to , we have converges to .

Continuity For Functions

Theorem: is continuous at such that .

Proof: (←): Given the above premise, let be a sequence such that . We want to prove . In other words, such that

such that by the premise.
such that for any since

At this point, depends on and depends on . , we found such that if , then , meaning .

(→): Proof by Contradiction.

The converse (ie. assuming the conclusion is false) is such that such that , such that but

Let . such that but
Let . such that but

Consider . We have but doesn’t converge to .

The converse: . is continuous at if and only if such that implies that .


Example 1

Prove that is not continous at .

Intuition: If we take an delta-neighborhood around , we will see there exists no points around it in the image.

Proof: Take . such that , but .


Example 2

Let . We want to show that it is continuous everywhere.

Proof: , , take . Then,


Example 3

Let . We want to show that it is continuous everywhere.

Proof: , , we want to find such that if , then .
Notice that .

Take . We get , and .
Then, .

Alternatively, take . We get , and . This gives us .


Remark

typically depends on both and .


Continuity For General Spaces

Theorem: is continuous if and only if open, is open. Remark. is not a function; it is defined as .

Proof:

(→) continuous, open. We want to show that is open, ie. such that .
Since is open, such that .
Since is continuous, we can find a such that if , then , which implies that , and therefore .

(←) Given that open, is open, , we want to show that such that .
is open. By our premise, is open.
Therefore, such that , hence if , then .