111. Extreme Value Theorem
The Extreme Value Theorem states that if a function is continuous on a closed interval , then must attain both a maximum and a minimum value within that interval. This means there exist points such that:
- Continuity: The function must be continuous on . Discontinuities (jumps, asymptotes, holes) can prevent the function from attaining an extreme value
Example
- Closed Interval: If the function is defined on an open interval , an extremum may not exist
Example
on has no maximum because it keeps increasing as .