1. 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

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

3. 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.

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

5. 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:

6. Intermediate Value Theorem 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.

7. 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. F21: solve

8. This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition. Graphing calculators can sometimes make non- continuous functions appear continuous. Graph: CATALOG F floor( Note resolution. The calculator “connects the dots” which covers up the discontinuities.

9. Graphing calculators can make non-continuous functions appear continuous. Graph: CATALOG F floor( GRAPH The open and closed circles do not show, but we can see the discontinuities.  If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph.