That g is uniformly 234 continuous on the space y

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Unformatted text preview: the function f is uniformly continuous at the point u we choose a number   0 such that whenever x X and d u, x   we have d f t ,f x  . 2 Now since u is a limit point of S we know that infinitely many members of S must lie in the ball B u, /2 and, using this fact, we choose a positive integer n  2/ such that either t n or x n lies in the ball B u, /2 . Since one of the points t n and x n lies within a distance /2 or u and since d tn, xn  1   n 2 we know that both t n and x n lie within a distance  of u. Thus d f tn , f u  d f u , f xn d f tn , f xn    2 2 which contradicts the way in which the points t n and x n were chosen. f. Suppose that S is a compressed subset of a metric space X. Prove that it is possible to find two sequences a n and b n in the set S such that an n Z bn n Z  and such that the inequality d a n , b n  1/n holds for every n Z  . Solution: Using the fact that S is compressed we choose two members of S, that we shall call a 1 and b 1 such that a 1 know that the set b 1 and d a 1 , b 1  1. Since S is compressed and the set a 1 , b 1 is finite we S a1, b1 is compressed. Using this fact we choose two members that we shall call a 2 and b 2 of the set S a 1 , b 1 such that a 2 b 2 and d a2, b2  1 . 2 By continuing this process we arrive at the desired sequences a n and b n . 237 g. Suppose that A and B are subsets of a metric space X, that A Þ B has no limit point, that A B  and that for every positive number  it is possible to find a member a of the set A and a member b of the set B such that d a, b  . Prove that there exists a continuous function f from X to R such that f is not uniformly continuous on X. Hint: Use Urysohn’s lemma. Since A Þ B has no limit point, the two sets A and B are closed. The continuous function A A  B fails to be uniformly continuous on X. h. Prove that if every continuous function from a given metric space X is uniformly continuous then the space X must be strongly complete. This part follows at once from parts f and g. 9 Differentiation Exercises on Derivatives 1. Given that f x  |x | for every number x, prove that f 0 does not exist. Since ft f0 lim  lim t  1 t0 t 0 t 0 t and ft f0  lim t  1 lim t0 t0 t 0 t the two sided limit ft f0 lim t0 t0 fails to exist. 2. Given that f x  |x | for all x 2, 1 Þ 0, 1 , prove that f 0 does exist. Hint: Observe that whenever x 2, 1 Þ 0, 1 and |x 0 |  1 we have x 0 and therefore, if x we have fx x f0 0 x x 0  1. 0 3. Given that f x  x|x | for every number x, determine whether or not f 0 exists. We observe that ft f0 t|t | lim  lim  lim|t |  0 t0 t0 t0 t t0 and so f 0  0. Students may want to sketch the graph of this function using Scientific Notebook. x|x | 238 0 4. This exercise concerns the function f defined by the equation x 2 sin fx  1 x if x 0 if x  0 0 . You should assume all the standard formulas for the derivatives of the functions sin and cos. a. Ask Scientific Notebook to make a 2D plot of the expressio...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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