Continuity Theorems

Extreme Value Theorem

If is continuous on , has a maximum and a minimum in .

Proof Hint

Proof is quite hard.

Intermediate Value Theorem

Let is continuous on . If such that or : such that .

Proof Hint

• Define
• Define
• Show that () exists. Assume and contradict these cases:
  • (use )
  • (use )
  • then contradict:
    • (similar to case)
    • (similar to case)

Sandwich (or Squeeze) Theorem

Let:

• For some :

Then .

Note

Works for any kind of x limits.

”No sudden changes”

Positive

Let be continuous on and

Proof Hint

Take

Negative

Let be continuous on and

Proof Hint

Take