1873_solutions

# F x y y and y x we see that f x y x y

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Unformatted text preview: ction f does not have a limit at the number a. Since x must be a limit point of each of the sets Ý, a S and S a, Ý , the desired result follows at once from Theorem 8.3.2. Some Exercises on Continuity 1. In each of the following cases, determine whether or not the function f is continuous at 0, 0 a. We define 218 x2y f x, y  if 0 x, y if x 2 y 2 0, 0 x, y  0, 0 . 2 0 -2 -4 x 0 -2 y 0 2 4 4 Since |f x, y | |y | for each point x, y we see that f x, y continuous at 0, 0 . 0 as x, y 0, 0 and so f is b. We define x2y f x, y  0 if x, y if x 4 y 2 0, 0 x, y  0, 0 . -4 0.5 x 0 2 -0.5 4 -4 y 2 4 We saw in an earlier exercise that the function f has no limit at 0, 0 and therefore f can’t be continuous at 0, 0 . c. We define x4y4 f x, y  0 if x, y if x 6 y 6 0, 0 x, y  0, 0 . 12 10 8 6 4 2 0 -5 y 5 4 2 0 x -4 Solution: From the inequality a2  b2 2 2 that holds for all real numbers a and b we observe that whenever x, y |ab | 219 0, 0 we have |x 3 y 3 | |xy | . 2 x6  y6 The continuity of the function f at 0, 0 now follows simply from the sandwich theorem. |f x, y |  |xy | 2. Given that X is a metric space, that a for every point x X and that f x  d a, x X, prove that the function f is continuous on X. Hint: First make the observation that whenever x and u belong to the space X we have |f x f u | d x, u . This inequality has come up several times already. To show that f is continuous at a given point x X, suppose that  0. We define   and observe that whenever u B x,  we have d f x ,f u d x, u  . 3. Suppose that f and g are functions from a metric space X into a metric space Y and that the inequality d f t ,f x d g t ,g x holds for all numbers t and x in S. Prove that f must be continuous at every point at which the function g is continuous. Suppose that x X and that g is continuous at x. To show that f is continuous at x, suppose that  0. Choose   0 such that, whenever t B x,  we have d g t ,g x  . Then, for every t B x,  we have d f t ,f x  . 4. Prove that if f is a continuous function from a metric space X into R k then the function f is continuous from X into R. Since fx | ft fx | ft whenever t and x belong to the space X, the continuity of f follows from Exercise 4. 5. Give an example of a continuous function f from a metric space X to a metric space Y and a closed subset H of X such that set f H fails to be closed in Y. We take X  R and Y  0, 1 and we define 1 fx  1  x2 for every x R. We see that f R  0, 1 which is not closed in Y. 6. Give an example of a continuous function f from a metric space X to a metric space Y and an open subset U of X such that set f U fails to be open in Y. We take X  0, 1 Þ 2, 3 and we define fx  x if 0 x 1 x 1 if 2 x 3. This function f is continuous from X to the metric space 0, 2 and, even though 0, 1 is an open subset of X, the set f 0, 1 fails to be open in the space 0, 2 . Of course we could have taken a discrete space for X. A challenge question would be to ask whether the student can co...
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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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