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Likewise show how, for the same subdivision, the lower sum for
f
may be used to construct a lower sum for
f
.
How close together can you be sure the upper and lower sums for
f
must be? For given
0, can you choose
n
suMciently large to
make2
?
And can you then choose a subdivision for which the difference
between the upper and lower sums for
f
(
b
a
)
? What then follows
about the difference between the upper and lower sums for
f
for this
subdivision of thedomain [
a
,
b
]? Deduce that
f
is integrable on
[
a
,
b
].
30 If, on thedomain [
a
,
b
], the sequence of integrable functions (
f
)
converges uniformly to the function
f
, usetheargument of qn 24 to
provethat thelimit as
n
of
f
is equal to
f
.
31 Let (
f
) be a sequence of functions which are integrable on the
domain [
a
,
b
] and which converge uniformly. Use an argument like
that of qn 24 to prove that the sequence of functions (
F
) defined
by
F
(
x
)
f
is uniformly convergent on the domain [
a
,
b
].
Summary
-
Uniform convergence, continuity and integration
Definition
(
f
) is a sequence of real functions,
f
:
A
R
, all
qns 5, 6
of which havethesamedomain
A
. If for each
x
(
x
) exists, then a
function
f
:
A
R may be defined by
f
(
x
)
lim
A
, thelimit lim
f
(
x
). This
f
is called the
pointwise
limit function
of the sequence (
f
f
).
Definition
The sequence of functions, (
f
), has thepointwise
qns 14, 15
limit function
f
:
A
R
.
If (sup
f
(
x
)
f
(
x
)
:
x
A
)
0as
n
,we
say that the sequence (
f
)
converges uniformly
to
f
.
Theorem
If the sequence of functions (
f
) converges
qn 20
uniformly to thefunction
f
:
A
R and each of
thefunctions
f
is continuous on its domain
A
,
then
f
is continuous on
A
.
Theorem
If the sequence of functions (
f
) converges
qns 29, 30
uniformly to thefunction
f
:[
a
,
b
]
R and each
of thefunctions
f
is integrable on [
a
,
b
] then
f
is
integrable on [
a
,
b
] and lim
f
f
.
