Intermediate Value Theorem

The Intermediate Value Theorem. If is a continuous function on the interval , and if is any value between and , then for some in This applet has several functions to use for exploring the intermediate value theorem. The "next" button changes the function. You can move the points and . A slider then moves between and . If a value can be found with it is shown.
Explore the various functions with and set where is continuous or not continuous. Are values found if is not continuous? Are there functions and boundaries where multiple values could work for ? Note the statement when no value is found. Is this logically correct?