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In-Depth Information
Uniform convergence and integration
We have shown that the uniform limit of a sequence of continuous
functions is continuous. Because of qn 10.39 this allows us to claim in
certain cases that the uniform limit of a sequence of integrable functions
is integrable.
24 What further condition must the functions of qn 20 satisfy if their
integrability is to be claimed?
Even when the domain
A
of thefunctions in qn 20 is a closed
interval, it is still an open question as to whether the limit of the
integrals is equal to the integral of the limit function.
Usetheinequality
f
f
(
f
f
)
f
f
(from qn
10.35), to provethat
lim
f
f
, or lim
f
(lim
f
).
We have shown that the limit of an integral is equal to the
integral of the limit in a context of uniform convergence but have
not shown that there was any need for uniform convergence for this
result.
25 Let
f
(
x
)
nx
(1
x
)
on thedomain [0, 1].
Verify that the pointwise limit function is the zero function
throughout this domain.
Provethat
f
n
/(2
n
2) on [0, 1] and deduce that
lim
f
lim
f
.
Is the convergence of the sequence (
f
) uniform? Usequstion 24.
Usecomputr softwareto draw thegraphs of somefunctions from
this sequence.
What is lim
sup
f
(
x
)
f
(
x
)
:0
x
1
?
26 Show that thersult of qn 24 may
not
be extended to improper
integrals by attempting to apply it to the sequence of functions
defined by
(
n
x
)/
n
when
x
n
,
n
],
[
f
(
x
)
0
otherwise.
