Page 227 - DMTH401_REAL ANALYSIS

Unit 17: Uniform Convergence and Differentiability

In Figure 17.5, uniform approximation of continuous function f (in blue) by polynomial (in red) Notes
in e-tube around f (in green).

Self Assessment

Fill in the blanks:

¥
¥
}
1. Let I Ì  and {g n n= 1 be a sequence of real-valued functions on I. Then å g is a
n
n = 1
..................................
2. Let { }f ¥ be a sequence of real-valued functions on I Ì  Then ................. is called a
.
n n= 1
uniform Cauchy sequence if:
$
"e > 0 N Î " , n m ³ N : f - f m sup < . e
n
¥
¥
3. Let { }f n n= 1 be a sequence of real-valued functions on I Ì  Then { }f n n= 1 ..................................
.
¥
of I if and only if { }f n n= 1 is a uniform Cauchy sequence on I.
n
4. Let å ¥ a x be a power series with a ....................... R. Then for any 0 £ < R the series
r
n= 0 n
r
converges uniformly on [ r- , ].
5. There is a function f :  ®  which is continuous but nowhere ..................................
17.6 Summary
 Let { }f ¥ be a sequence of real-valued function on [ , ].a b If (a) all f are differentiable,
n n= 1 n
b
¢
,
(b)’ all f are continuous, (c) f ® uniformly, for some function h :[ , ]®  (d)
a
¢
h
n n
b
,
a
a
c
b
c $ Î [ , ] s.t. f n ( ) converges then f converges uniformly to some f :[ , ] ®  in
n
x
addition, this uniform limit f is differentiable and f ¢ ( ) h ( ).
x
=
¥
}
 Let I Ì  and {g n n= 1 be a sequence of real-valued functions on I. Then
¥
å g n
n= 1
is a series of functions.
n
r
 Let å ¥ a x be a power series with a radius of convergence R. Then for any 0 £ < R the
n= 0 n
r
series converges uniformly on [ r- , ].
 There is a function f :  ®  which is continuous but nowhere differentiable.
17.7 Keywords
Series of Functions: Let I Ì  and {g } ¥ be a sequence of real-valued functions on I. Then
n n= 1
¥
å g n
n= 1
is a series of functions.
Continuity for Series: If (a) all g are are continuous and (b) å ¥ g n converges uniformly then
n n= 1
å ¥ n= 1 g is continuous.
n
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