1. Continuity
Sec. 2.3

2. Warm-up 9/3/2013
a. Find the average rate of change on the interval [2 , 5].
b. Find the instantaneous rate of change at x = 2.

3. This function has discontinuities at x=1 and x=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. 1 2 3 4
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

4. Removable Discontinuities:
(You can fill the hole.)
Non-Removable (Essential) Discontinuities:
infinite
oscillating
jump

5. 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.

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

7. 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:

8. 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

9. Is any real number exactly one less than its cube?
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.

10. Graphing calculators can sometimes make non-continuous functions appear continuous.
Note resolution.
The calculator “connects the dots” which covers up the discontinuities.

11. Graphing calculators can make non-continuous functions appear continuous.
If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct graph.
The open and closed circles do not show, but we can see the discontinuities. p