1. Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities that functions may contain. 3.Be able to determine if a function is continuous on a closed interval. 4.Be able to determine one-sided limits and continuity on a closed interval. Critical Vocabulary: Limit, Continuous, Continuity, Composite Function

2. I. Continuity Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c This means a graph will contain no _____, _____, or _____ Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.

3. I. Continuity What Causes discontinuity? 1. The function is not defined at c. This is an example of a ____ in the graph at ___ Concept: The function is not defined at c. __________________ Let’s look at at f(x) = ½x - 2

4. I. Continuity What Causes discontinuity? 2. The limit of f(x) does not exist at x = c This is an example of a ____ in the graph at ______ Concept: The limit does not exist at x = c Let’s look at at

5. I. Continuity What Causes discontinuity? 3. The limit of f(x) exists at x = c but is not equal to f(c). This is an example of a _____ in the graph Concept: The behavior (limit) and where its defined (f(c)) are __________________ Let’s look at the first graph again What is the limit as x approaches -2? What is f(-2)?

6. A function f is continuous at c if the following three conditions are met: 1. ___________________________ 2. ___________________________ 3. ___________________________ I. Continuity Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c This means a graph will contain no HOLES, JUMPS, or GAPS Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.

7. Objectives: 1.Be able to define continuity by determine if a graph is continuous. 2.Be able to identify and find the different types of discontinuities that functions may contain. 3.Be able to determine if a function is continuous on a closed interval. 4.Be able to determine one-sided limits and continuity on a closed interval. Critical Vocabulary: Limit, Continuous, Continuity, Composite Function

8. II. Discontinuities When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous. Discontinuity is broken into 2 Categories: 1. Removable: A discontinuity is removable if you COULD define f(c).

9. II. Discontinuities When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous. Discontinuity is broken into 2 Categories: 1. Removable: A discontinuity is removable if you COULD define f(c). 2. Non-Removable: A discontinuity is non-removable if you CANNOT define f(c).

10. II. Discontinuities When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous. Discontinuity is broken into 2 Categories: 1. Removable: A discontinuity is removable if you COULD define f(c). 2. Non-Removable: A discontinuity is non-removable if you CANNOT define f(c). Example 1:What is the Domain?

11. II. Discontinuities When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous. Discontinuity is broken into 2 Categories: A discontinuity is removable if you COULD define f(c). A discontinuity is non-removable if you CANNOT define f(c). Example 2:What is the Domain?

12. II. Discontinuities When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous. Discontinuity is broken into 2 Categories: 1. Removable: A discontinuity is removable if you COULD define f(c). 2. Non-Removable: A discontinuity is non-removable if you CANNOT define f(c). Example 3: Discuss the continuity of the composite function f(g(x))

13. II. Discontinuities Example 5: Graph the piecewise function, then determine on which intervals the graph is continuous.

14. III. Closed Intervals Example 5: Discuss the continuity on the closed interval. Closed Interval: Focusing on specific portion (domian) of a graph. [a, b]

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16. Objectives: 1.Be able to define continuity by determine if a graph is continuous. 2.Be able to identify and find the different types of discontinuities that functions may contain. 3.Be able to determine if a function is continuous on a closed interval. 4.Be able to determine one-sided limits and continuity on a closed interval. Critical Vocabulary: Limit, Continuous, Continuity, Composite Function

17. IV. One-Sided Limits What does a One-Sided look like? c cApproach from the right only cApproach from the left only

18. IV. One-Sided Limits Example 1: Graph then find the limits What’s the domain? x f(x)

19. IV. One-Sided Limits Example 1: Graph then find the limits x f(x) Is this graph continuous?

20. 1.Page 237 #23-43 odd, 49-55 odd, 61, 63, 77 2. Page 236 #1-17 odd, 79, 88