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18.100B.Homework8

18.100B.Homework8 - be a continuous function and assume...

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18.100B/C: Fall 2008 Homework 8 Available Tuesday, October 28 Due Wednesday, November 5 Turn in the homework by 11am on Wednesday, November 5th, in 2-108. For 18.100B it should be put in the bin corresponding to the lecture you regularly attend (regardless of which one you may be officially enrolled in). If you are enrolled in 18.100C put your homework in the C bin in any case. 1. (a) (10 pts) Assume that f : R -→ R and for any x, y R | f ( x ) - f ( y ) | ≤ C | x - y | α for some C > 0 and α > 0 . Prove that if α > 1 then f is constant. [ Hint: What is the derivative of a constant function?] (b) (5pts) If α 1 , is f necessarily differentiable? 2. (a) (5pts) Assume f : (0 , 1] -→ R is differentiable and | f ( x ) | ≤ M for all x (0 , 1] . Define the sequence a n = f (1 /n ) and prove that a n converges. (b) (10pts) Let f : R -→
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Unformatted text preview: be a continuous function and assume that f ± ( x ) exists ev-erywhere. Assume also that there is a constant α < 1 such that, for any x ∈ R , | f ± ( x ) | ≤ α < 1 . Prove that f has a unique fixed point. [ Hint : use problem 7 from Homework 6.] (c) (5pts) Suppose we only know that | f ± ( x ) | < 1 for all x . Does the conclusion of part (b) still hold? [ Hint : think of slant asymptotes.] 3. (10 pts) Problem # 2 page 114 in Rudin . 4. (20 pts) Problem # 13, parts a), b), c), and d), page 115 in Rudin . 5. (15 pts) Problem # 6 page 114 in Rudin . 6. (10 pts) Problem # 7 page 114 in Rudin . 7. (10 pts) Problem # 26 page 119 in Rudin ....
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