7.3 Uniform convergence
109
Example 7.3.2.
Let
f
n
(
x
) =
x
n
on
I
= [0
,
1]. Then
{
f
n
}
converges pointwise and
f
(
x
) =
0
,
0
≤
x <
1
,
1
,
x
= 1
.
So a sequence of continuous functions can converge pointwise to something which is not
continuous! In fact,
f
n
∈
C
∞
(
I
), but
f /
∈
C
(
I
)!
Even worse:
Example 7.3.3.
Let
f
k
(
x
) = lim
n
→∞
(cos
k
!
xπ
)
2
n
.
Then whenever
k
!
x
is an integer,
f
k
(
x
) = 1. If
x
=
p/q
is rational, then for
k
≥
q
,
f
n
(
x
) = 1. If
k
!
x
is not an integer (for
example, if
x
is irrational), then
f
k
(
x
) = 0. We have an everywhere discontinuous limit
function
f
(
x
) = lim
k
→∞
lim
n
→∞
(cos
k
!
xπ
)
2
n
=
0
,
x
∈
R
\
Q
,
1
,
x
∈
Q
.
Example 7.3.4.
Let
f
n
(
x
) =
x
2
(1+
x
2
)
n
on
R
and consider
f
(
x
) =
∞
X
n
=0
f
n
(
x
) =
∞
X
n
=0
x
2
(1 +
x
2
)
n
=
0
,
x
= 0
,
1 +
x
2
,
x
6
= 0
,
since the series is geometric for
x
6
= 0. So a series of continuous functions can converge
pointwise to something which is not continuous! (Not even integrable!)
Example 7.3.5.
Let
f
n
(
x
) =
sin
nx
√
n
on
R
. Then
f
(
x
) = lim
n
→∞
f
n
(
x
) = 0
,
∀
x
∈
R
,
so
f
0
(
x
) = 0. On the other hand,
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 Fall '11
 Wong
 Calculus, lim, FN, Uniform convergence, Pointwise convergence, limn→∞ fn

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