lecture9 - Lecture 9: Continuity Chapter 2, Section 9 Def....

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Lecture 9: Continuity Chapter 2, Section 9 Def. A function f is continuous at a number a if lim x ! a f ( x ) = f ( a ) : If f is deflned on an open interval including x = a but is not continuous there, then f is discontinuous at a . The deflnition implies three conditions for continuity: 1. 2. 3. Note that f is continuous at x = a if the limit as x ! a gives the same number as evaluating the function at x = a .
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ex. Consider the following graph of a function f ( x ): 6 - ? ± At which numbers is f discontinuous? Can we deflne or redeflne f ( x ) to make it continuous at any of those values?
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We classify two types of discontinuities at a point x = a . Removable Nonremovable Jump Discontinuity Inflnite Discontinuity
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Functions that are continuous The following familiar functions are continuous for each x in their domain: Polynomials Rational functions Root functions Trigonometric and Inverse Trigonometric functions Exponential functions Logarithmic functions The limit laws can be used to verify the following Theorem: If functions f and g
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lecture9 - Lecture 9: Continuity Chapter 2, Section 9 Def....

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