1. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 1 What you ’ ll learn about Definition of continuity at a point Types of discontinuities Sums, differences, products, quotients, and compositions of continuous functions Common continuous functions Continuity and the Intermediate Value Theorem …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.

2. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 2 Continuity at a Point

3. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 3 Example Continuity at a Point o

4. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 4 You Try For which intervals is the function continuous? Where is the function discontinuous? Slide 2.3- 4

5. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 5 Continuity at a Point

6. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 6 Continuity at a Point

7. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 7 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)

8. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 8 Continuity at a Point

9. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 9 Example Continuity at a Point [  5,5] by [  5,10]

10. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 10 Continuous Functions

11. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 11 Continuous Functions [  5,5] by [  5,10]

12. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 12 Properties of Continuous Functions

13. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 13 Composite of Continuous Functions

14. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 14 Intermediate Value Theorem for Continuous Functions

15. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 15 Intermediate Value Theorem for Continuous Functions

16. Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 16 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. Intermediate Value Theorem for Continuous Functions