Chapter4 - Chapter 4: Functional Limits and Continuity...

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Unformatted text preview: Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue Chapter 4: Functional Limits and Continuity Peter W. White white@tarleton.edu Initial development by Keith E. Emmert Department of Mathematics Tarleton State University Fall 2011 / Real Anaylsis I Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue Overview Discussion: Examples of Dirichlet and Thomae Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The Intermediate Value Theorem Sets of Discontinuity Epilogue Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue Dirichlets Function Example 1 Let f ( x ) = ( 1 , if x Q , if x 6 Q . This function is nowhere continuous on R . (Trust me.) Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue Modified Dirichlets Function Example 2 Let f ( x ) = ( x , if x Q , if x 6 Q . This function is continuous only at c = 0. (Trust me.) Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue Thomaes Function Example 3 Let f ( x ) = 1 , if x = 1 n , if x = m n Q \{ } is in lowest terms with n > , if x 6 Q . This function is continuous on the irrationals. It is discontinuous at the rationals. (Trust me.) Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue Overview Discussion: Examples of Dirichlet and Thomae Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The Intermediate Value Theorem Sets of Discontinuity Epilogue Chapter 4: Functional Limits and Continuity PWhite Discussion Functional Limits Combinations of Continuous Functions Continuous Functions on Compact Sets The IVT Sets of Discontinuity Epilogue- Version of Limit of a Function Remark 4 Recall from Chapter 3: I Definition: A point x is a limit point of a set A if every- neighborhood V ( x ) of x intersects the set A in some point other than x. I Theorem: A point x is a limit point of a set A if and only if x = lim a n for some sequence ( a n ) contained in A satisfying a n 6 = x for all n N ....
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Chapter4 - Chapter 4: Functional Limits and Continuity...

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