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U n . Using the fact that the
function f n is continuous at the number x, choose 2 0 such that the inequality
f n t f n x 
3
holds whenever t x  2 . We now define to be the smaller of the the numbers 1 and 2 .
Whenever t x  we have
f t f x  f t f n t  f n t f n x  f n x f x .
Now, whenever t x  , the fact that both t and x belong to the set U n gives us
f t f n t  limf j t f n t 
jÝ
3
and
f x f n x  limf j x f n x 
jÝ
3
and we conclude that f t f x 
.
6. Suppose that f n is a sequence of continuous functions on an interval a, b where a b and that f n
converges pointwise on a, b to a function f. Suppose that for every positive number , the set V introduced
in Exercise 4 is now called V . Apply Exercise 2 to the sequence of sets
a, b
V1
n
and deduce that if 367 D Ý V
n1 1
n then D is a dense subset of the interval a, b . Prove that the function f is continuous at every number in that
belongs to this dense set D.
From Exercise 5a we know that each of the set V 1 is an open dense subset of a, b and it
n
follows from Exercise 3 that the set
D Ý V
n1 1
n a, b Ý
n1 R V1
n is dense in a, b . To see that f is continuous on D, suppose that x D and that 0. Choose a
positive integer n such that 1/n , and, using Exercise 5b, and the fact that x V 1 , choose
n
0 such that, whenever t x  we have
1.
f t f x 
n Some Exercises on the Properties of Uniform
Convergence
1. Determine whether the following statement is true or false:
If f n is uniformly continuous on a set S for every positive integer n and if the sequence f n converges
uniformly on S to a function f then f must be uniformly continuous on S.
The statement is true. Suppose that 0. Using the fact that f n f uniformly on S, choose an
integer N such that the inequality
supf n f 
3
holds whenver n N. Using the fact that the function f N is uniformly continuous on S, choose 0
such that the inequality
f N t f N x 
3
holds whenever t and x belong to S and t x  . Then, whenever t and x belong to S and
t x  we have
f t f x  f t f N t f N t f N x f N x f x 
f t f N t  f N t f N x  f N x fx  . 2. Determine whether the following statement is true or false:
If f n is uniformly continuous on a set S for infinitely many positive integers n and if the sequence f n
converges uniformly on S to a function f then f must be uniformly continuous on S.
The statement is true and the proof is nearly identical to the one used in Exercise 1. The only
difference is that, instead of using the uniform continuity of the function f N we have to choose an
integer n N such that the function f n is uniformly continuous.
3. A family of functions is said to be equicontinuous on a set S if for every 0 and every number x S
there exists a number 0 such that whenever f
and whenever t lies in the set S
x , x we
have
f t f x  .
Prove that if a sequence f n converges uniformly on S and if each functi...
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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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