Page 208 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
=
x
Example: If the interval is not closed. Let I = (0,1) and f n ( ) x n .
(a) f  0 pointwise
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(b) x is continuous.
n
x
(c) f ( ) = 0 is continuous
f
"
(d) { } is monotonic x Î (0, 1).
n
But f does not converge to ( ) 0f x = uniformly.
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Self Assessment

Fill in the blanks:
1. If there is no pointwise convergence, there is no ..............................
2. If f ............................ to some f test the uniform convergence to f.
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3. A collection of open intervals { }I a a Î A is a cover of a set ..........................

I
4. Given a set S and its cover { }I , a ................................. is a subcollection of { } , which
a aÎA a aÎA
itself is a cover for S.
15.5 Summary

 Let f f f … be a sequence of functions from one metric space into another, such that for
1 2 3
any x in the domain, the images f (x) f (x) f (x) … form a convergent sequence. Let g(x) be
1 2 3
the limit of this sequence. Thus the function g is the limit of the functions f f f etc.
1 2 3
 This sequence of functions is uniformly convergent throughout a region R if, for every 
there is n such that fj(x) is within e(g(x), for every x in R and for every j ³ n. The functions
all approach g(R) together, one n fits all. This is similar to uniform continuity, where one
d fits all.

 If the range space is complex, or a real vector space, the sequence of functions is uniformly
convergent iff. All the component functions are uniformly convergent. Given an e, find f
n
that is close to g, and the components of f must be close to the components of g, for all x.
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Conversely, if the components are within e then the n dimensional function is within ne,
for all x.
 Without uniform convergence, g is rather unpredictable. Let the domain be the closed
interval [0, 1], and let f = x . Note that the sequence f approaches a function g that is
n n
identically 0, except for g(1) = 1. Each function in the sequence is uniformly continuous, yet
the limit function isn’t even continuous.

15.6 Keywords

Heine-Borel: Every cover of closed interval [ , ]a b has a finite subcover.

Dini’s Theorem: Let { }f ¥ be a sequence of real-valued functions on [ , ].a b If:
n n= 1
a
f
x
(a) f  f pointwise, (b) all f are continuous, (c) f is continuous and (d) x" Î [ , ] : { ( )} ¥ n= 1
b
n
n
n
is monotone then f  f uniformly.
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