Sample Exam 2D

Sample Exam 2D - f is a bounded set 5 Let f be a continuous...

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MATH 311 EXAM 2D Directions for Part I: For each of the following, provide the appropriate definition or theorem. 1. Accumulation Point 2. Functional Limit 3. Monotone Function 4. Continuity at a Point 5. Uniform Continuity 6. Open Set and Closed Set 7. Compact Set 8. Heine-Borel Theorem 9. Extreme Value Theorem 10. Intermediate Value Theorem 11. *1 st Continuous Function Theorem 12. *2 nd Continuous Function Theorem 13. *4 th Continuous Function Theorem 14. *Sequential Compactness Theorem Directions for Part II: Complete the following problems. If a problem asks for a proof, make explicit what you are assuming. Make sure you justify each step. Good luck! 1. Prove that every open interval is open and every closed interval is closed. 2. Verify that . 3. Suppose is uniformly continuous and is a Cauchy sequence in D . Prove that is a Cauchy sequence. 4. Suppose E is bounded and is uniformly continuous. Prove that , the range of
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Unformatted text preview: f , is a bounded set. 5. Let f be a continuous function on R and let r be a real number. Prove that the set is open. 6. Consider a function . Prove that f is continuous if and only if for every open set , it follows that is open. Prove that if f is continuous and E is a closed set, then is closed. 7. Let be a family of nonempty compact sets in R , i.e. is compact for all . a. Show that if is empty, then there is some positive integer N such that is empty. b. Now suppose . Show that if for each n , then is nonempty ( Hint: Use part a.) 8. Let f be a function defined on all of R . Show that if there exists a constant c such that and , then f has a fixed point , that is, a point p such that . ( Hint: Fix a point in R and show that the sequence given by is Cauchy. )...
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This note was uploaded on 12/13/2011 for the course MATH 115 taught by Professor Plotkin during the Spring '08 term at Rutgers.

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