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
• Start with
• Define
• Show that () exists.
• Use the continuity definition. Assume and contradict these cases:
  • (use )
  • (use )
  • then contradict:
    • (similar to case)
    • (similar to case)

Sandwich (or Squeeze) Theorem

Let:

• For some :

Then .

Works for any kind of x limits.

Proof Hint

• Consider the limit definition of both and at
• Show that is bounded below by
• Show that is bounded above by
• That concludes is bounded by which leads to the required limit

”No sudden changes”

Positive

Let be continuous on and

Proof Hint

Take

Negative

Let be continuous on and

Proof Hint

Take