1. 2.3 Continuity Grand Canyon, Arizona Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

2. Methods for Evaluating Limits Direct Substitution Algebraic Techniques –Factor –Rationalize Denom. –Multiply by Conjugate Sandwich Theorem Limits at infinity

3. 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

4. How to show continuity 1)Limit of F(x) as x  c exist 2)F(c) exists 3)Limit of F(x) = F(c)

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

6. 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.

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

8. 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:

9. 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.

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. menu 3 Algebra 1 Solve

11. Now press to ask the calculator for an approximate answer, and notice how much quicker it is. enter ctrl We will find the same command another way.  Functions, commands and tools on the calculator can often be found in more than one way. If you can’t find something, look in the catalog. S solve( If you press it takes the calculator a while to find the answer, because it is trying to find an exact answer first. enter Use the up arrow to highlight the formula, and press to make another copy. enter