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lecture9 - 4 3 2 1 0 4 3 2 1 1 0 1 2 3 4 2 3 4 4 3 2 1 0 4...

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Lecture 9: Continuity (Section 2.5) Def. A function f is continuous at a number a if lim x ! a f ( x ) = f ( a ) : If f is deflned on an interval near x = a but is not continuous there, then f is discontinuous at a . The deflnition implies three conditions for continuity: 1. 2. 3. On an interval for which a function is continuous its graph has no jumps, holes, or gaps.
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There are two types of discontinuities at a point x = a . Removable Nonremovable Jump Discontinuity Inflnite Discontinuity
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ex. Consider the following graph: 6 - ? At which numbers is f discontinuous? What type of discontinuity? Can we deflne or redeflne f ( x ) to make it continuous at any of those values?
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ex. Discuss the continuity of f ( x ) = x 2 ¡ 1 x 3 ¡ x . What types of discontinuities does f ( x ) have? Could we deflne f ( x ) to make it continuous at any of those discontinuities? 6 - ?
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ex. Discuss continuity and determine any removable or nonremovable discontinuities for f ( x ) = 8 > < > : x ¡ 1 if x < 0 j x j + 1 if 0 x 1 2 p x if x > 1 -2 -3 4 2 3 0 -2 1 -4 -4 4 1 -3 -1 3 -1 2 0
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Def. A function f is continuous from the right at x = a if f is continuous from the left at a if A function is continuous on an interval if
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ex.
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