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Determine the pointwise limit function of the sequence (
f
).
Prove that the convergence is uniform, and evaluate
f
and
f
on thedomain R.
If we ask whether convergent sequences of integrable functions
which are not necessarily continuous must converge to an integrable
limit we would have part of an answer if we could construct a sequence
of integrable functions which converge to Dirichlet's function given by
1
0
when
x
is rational, and
when
x
is irrational,
f
(
x
)
since this is an example of a function which is not Riemann integrable
on any interval.
27 We construct a sequence of functions on the closed interval [
a
,
b
].
There is a countable infinity of rational numbers in this interval so
there is a sequence in [
a
,
b
], which wedenoteby
x
,
x
,
x
,...,
x
,...,
which contains them all.
On the closed interval [
a
,
b
] wedefinethefunction
f
by
1
0
when
x
x
,
x
,...,
x
, and
f
(
x
)
otherwise.
Illustrate
f
on a graph. Are these functions integrable? Is
every member of the sequence (
f
,
f
and
f
) integrable?
What is thepointwiselimit function?
28 In qn 27, evaluate
f
)
.
Determine whether the convergence of the sequence in qn 27 is
uniform.
(
x
)
f
(
x
Having found that thepointwiselimit of integrablefunctions may
not be integrable, we now ask whether the uniform limit of a sequence
of integrable functions must be integrable.
29 On thedomain [
a
,
b
]welet(
f
) be a sequence of integrable
functions which tends uniformly to the function
f
. Thecondition of
uniform convergence, that for suMciently large
n
,
f
(
x
)
f
(
x
)
,
for any positive
, allows us to set an upper bound and a lower
bound on thefunction
f
, since
f
, for all
x
.
Now for any subdivision of the domain whatsoever, show how an
upper sum for
f
(
x
)
f
(
x
)
f
(
x
)
may be used to construct an upper sum for
f
.
