Properties of Continuous Functions

Extreme Value Theorem and Intermediate Value Theorem

1. Extreme Value Theorem
   If a function is continuous on a closed interval , then must have both a maximum and a minimum value on this interval.
   
   
2. Intermediate Value Theorem
1. Intermediate Value Theorem
   If a function is continuous on a closed interval and , then for any value between and , there exists at least one value between and such that .
   
   
2. Application of the Intermediate Value Theorem
   If a function is continuous on a closed interval and the signs of and are opposite, meaning , then there is at least one value between and such that .
   In other words, the equation has at least one real solution between and .