1. Continuity Grand Canyon, Arizona Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

2. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function 1234 1 2

3. jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the hole.)

4. Removing a discontinuity: has a discontinuity at. Write an extended function that is continuous at. Note: There is another discontinuity at that can not be removed.

5. Removing a discontinuity: Note: There is another discontinuity at that can not be removed.

6. Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. Also: Composites of continuous functions are continuous. examples:

7. Definition of Continuity Continuity at a point: A function f is continuous at c if the following three conditions are met A function is continuous on an open interval If it is continuous at each point in the interval. A function that is continuous on the entire real line is everywhere continuous

8. Formal Definition of Continuity on an Interval A function f is continuous on a closed interval If it is continuous on the open interval and The function f is continuous from the right at and continuous from the left at

9. Intermediate Value Theorem (IVT) If a function is continuous between a and b, then it takes on every value between and. Because the function is continuous, it must take on every y value between and.

10. Example 5: Is any real number exactly one less than its cube? (Note that this doesn’t ask what the number is, only if it exists.) Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. Use your calculator to find an approximate solution. or

11. Graphing calculators can make non-continuous functions appear continuous. The calculator “connects the dots” which covers up the discontinuities. Use your calculator to graph the function

12. Graphing calculators can make non-continuous functions appear continuous. Graph: GREATEST INTEGER FUNCTION

13. Graphing calculators can make non-continuous functions appear continuous. If we change the plot style to “dot”, we get a graph that is closer to the correct graph of the function. The open and closed circles do not show, but we can see the discontinuities. 