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.
- Spring '08