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Objectives: Be able to define continuity by determining if a graph is continuous.

Continuity and One-Sided Limits. Objectives: Be able to define continuity by determining if a graph is continuous. Be able to identify and find the different types of discontinuities that functions may contain. Be able to determine if a function is continuous on a closed interval.

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Objectives: Be able to define continuity by determining if a graph is continuous.

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

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

  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.

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

  5. I. Continuity What Causes discontinuity? 2. The limit of f(x) does not exist at x = c Let’s look at at This is an example of a gap in the graph at x = 3 Concept: The limit does not exist at x = c c

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

  7. 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. A function f is continuous at c if the following three conditions are met: 1. f(c) is defined 2. exists 3.

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

  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). c c HOLES JUMPS

  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). A discontinuity is non-removable if you CANNOT define f(c). 2. Non-Removable: c c GAPS ASYMPTOTES

  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: 1. Removable: A discontinuity is removable if you COULD define f(c). A discontinuity is non-removable if you CANNOT define f(c). 2. Non-Removable: Example 1: What is the Domain? Linear Function Has a Removable discontinuity at x = -1 Specific: Hole at (-1, -2) What intervals is the graph continuous?

  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). A discontinuity is non-removable if you CANNOT define f(c). 2. Non-Removable: Example 2: What is the Domain? Rational Function Has a Removable discontinuity at x = 3 Specific: Hole at (3, 1/6) Has a Non-Removable discontinuity at x = -3 What intervals is the graph continuous?

  13. 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). A discontinuity is non-removable if you CANNOT define f(c). 2. Non-Removable: Example 3: Discuss the continuity of the composite function f(g(x)) x + 1 > 0 x > -1 What intervals is the graph continuous?

  14. II. Discontinuities Example 5: Graph the piecewise function, then determine on which intervals the graph is continuous. Non-Removable discontinuity at x = 0 What intervals is the graph continuous?

  15. III. Closed Intervals Closed Interval: Focusing on specific portion (domian) of a graph. [a, b] Example 5: Discuss the continuity on the closed interval. Non-Removable discontinuity at x = 2 What intervals is the graph continuous?

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

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

  18. IV. One-Sided Limits What does a One-Sided look like? c c Approach from the right only c Approach from the left only Example 1: Graph

  19. IV. One-Sided Limits Example 1: Graph then find the limits What’s the domain? 0 2 0 DNE 0 2 0 DNE DNE DNE

  20. IV. One-Sided Limits Example 1: Graph then find the limits 3 3 4 0 3 3 3 Is this graph continuous? Has a Removable discontinuity at x = 3 Specific: Hole at (1, 3)

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

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