Graphics Reference
In-Depth Information
The critical question is whether you can use
n
to control the greatest
values of
f
(
x
)
f
(
x
)
.
18
(i) On [0,
a
],
f
(
x
)
f
(
x
)
a
0as
n
.
(ii) On [0, 1),
f
(
x
)
1as
x
1
,so
f
(
x
)
f
(
x
)
:0
x
1
.
This is a tricky and uncomfortable example, but it shows what a subtle
business uniform convergence may be.
(
x
)
f
(
x
)
1
sup
f
19 Thepointwiselimit function is given by
f
(
x
)
0.
f
f
(
x
)
:
x
R
sup
(
x
)
irrespective of
n
.
20 By the uniform convergence of (
f
)
f
,
(sup
f
(
x
)
f
(
x
)
:
x
A
)
0as
n
.
So for some
N
,
f
(
x
)
f
(
x
)
and
f
(
a
)
f
(
a
)
for
n
N
.
Sinceeach
f
is continuous a suitable
may befound.
Now, if
x
a
,
f
(
x
)
f
(
a
)
since we know the three inequalities
hold for somevalueof
n
. Thus lim
f
(
x
)
f
(
a
), and
f
is continuous at
a
.
22 First equality by continuity of
f
at
a
.
Second equality by definition of pointwise limit function.
Third equality by uniform convergence and qn 20.
Fourth equality by definition of pointwise limit function.
23 In qn 3,
lim
lim
f
(
x
)
lim
1
1.
lim
lim
f
(
x
)
lim
0
0.
24
A
[
a
,
b
]. For suMciently large
n
,
f
f
,
So
f
f
(
b
a
). Useqn 10.35.
25
lim
f
0.
f
(2
n
.
lim
f
(1/
1))
0.
n
sup
f
(
x
)
f
(
x
)
:0
x
1
(1
2
n
) ยท(1
1/(2
n
))
as
n
.
26 Thepointwiselimit function is given by
f
(
x
)
0.
f
1.
f
0.
27 Thefunction
f
is integrable as in qn 10.14. The pointwise limit function is
Dirichlet's function, qn 6.20.
