1873_solutions

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online