1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Continuity

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 2 Quick Review

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 3 Quick Review

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 Quick Review

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Quick Review Solutions

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Quick Review Solutions

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 10 What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions …and why Continuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 11 Continuity at a Point

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 12 Example Continuity at a Point o

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Continuity at a Point

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 14 Continuity at a Point If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 15 Continuity at a Point The typical discontinuity types are: a)Removable(2.21b and 2.21c) b)Jump(2.21d) c)Infinite(2.21e) d)Oscillating (2.21f)

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 Continuity at a Point

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 Example Continuity at a Point [-5,5] by [-5,10]

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 19 Continuous Functions [-5,5] by [-5,10]

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 20 Properties of Continuous Functions

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 21 Composite of Continuous Functions

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 22 Intermediate Value Theorem for Continuous Functions

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 23 Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.