Sample Exam 2B

Sample Exam 2B - and For example and Consider the function...

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MATH 311 EXAM 2B 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. Suppose satisfy and . Give a direct proof that . 2. Let denote the greatest integer function , i.e. such that
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Unformatted text preview: and . For example, and . Consider the function defined by . For what points does f have a limit? Prove your claim. 3. Suppose that (H1) f is a function which is defined and continuous on (H2) g is a function which is defined and continuous on (H3) Define a function h on by Give an proof that h is continuous at 0. 4. Prove directly that is not uniformly continuous on . 5. Prove that if K is compact, then K is closed and bounded. 6. For this problem, you may assume that if K is compact and is continuous, then is compact. Prove that if K is compact and is continuous, then achieves both its supremum and infimum, i.e. prove that f attains a maximum and minimum value....
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