- Definition: A function
*f*is**continuous at a number**if*a*

This definition implicitly requires three things if

*f*is continuous at*a*:1.

*f(a)*is defined (that is,*a*is in the domain of*f*)2. exists

3.

- If and are continuous functions with and

, find .

Use the definition of continuity and the properties of limits to show that the funtion is continous at the given number a. ,

- The Intermediate Value Theorem

Suppose that is continuous on the closed interval and let be any number between and , where . Then there exists a number in such that .~ ~ ~ Use the Intermediate Value Theorem to show that there is

a root of the given equation in the specified interval.