Limit of a Composite Function

1. Limit of a Composite Function If f and g are functions such that and , then

2. Example Because and it follows that

3. Definition of Continuity Continuity at a Point: A function f is continuous at cif the following three conditions are met. 1. f(c) is defined. 2. 3.

4. Continuity Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continuous at each point on the interval. A function that is continuous on the entire real line is everywhere continuous.

5. Continuity Continuity on a Closed Interval: A function f is continuous on a closed interval [a,b] if it is continuous on the open interval (a,b) and

6. Intermediate Value Theorem If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.