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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.
 Fall '08
 STAFF
 Math, Calculus

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