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

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

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