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Chapter 2 Section 4

Chapter 2 Section 4 - Chapter 2 Section 4 Continuity 1...

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1 Chapter 2 Section 4 Continuity
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2 Objectives: 1. Determine continuity of a function at a point. 2. Identify the three common types of discontinuities. 3. Rewrite a function with a removable discontinuity so that the function is continuous. 4. Use the continuity laws to determine whether a function is continuous. 5. Determine continuity at an endpoint. 6. Substitution method.
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3 Definition A function f is continuous at a number “c” if lim x c f(x) = f(c). 3 “implied” requirements 1. f(c) is defined 2. lim x c f(x) exists 3. lim x c f(x) = f(c) if any of the three fail, then the function is discontinuous at x = c If f(x) is continuous at all points in its domain, f is simply called continuous .
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4 Example: Show f(x) = x 2 - x is continuous at x = 4. 1. f(4) = 4 2 - 4 = 12 2. lim x 4 f(x) = lim x 4 (x 2 – x) = lim x 4 (x 2 ) – lim x 4 x = (lim x 4 x)·(lim x 4 x) – lim x 4 x = (4)·(4) – 4 = 12 3. lim x 4 f(x) = f(4) therefore, continuous at x = 4.
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There are three common types of discontinuities:
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