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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 xx  for every number x, determine whether or not f 0 exists.
We observe that
ft f0
tt 
lim
lim
limt  0
t0
t0
t0 t
t0
and so f 0 0. Students may want to sketch the graph of this function using Scientific Notebook.
xx  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.
 Fall '08
 STAFF
 Math, Calculus

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