Chapter4

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

This preview shows pages 1–8. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 [email protected] 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 Dirichlet’s 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 Dirichlet’s 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 Thomae’s 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- δ 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 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 48

Chapter4 - Chapter 4 Functional Limits and Continuity...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online