Sample Exam 2A

Sample Exam 2A - explicit what you are assuming Make sure...

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MATH 311 EXAM 2A 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
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Unformatted text preview: explicit what you are assuming. Make sure you justify each step. Good luck! 1. Let for all x in R . Verify that 2. Show that given by is uniformly continuous. 3. Suppose is continuous. Let p be a real number. Prove the set is open. 4. Suppose are compact sets in R . Prove that is compact. (You may prove this by any method, but you should know how to prove it directly.) 5. Give an example of an open cover for that has no finite subcover. 6. Suppose is continuous. Prove there exists a point p in such that ....
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