Graphics Reference
In-Depth Information
36 For unbounded domains use the functions of qn 8.
For bounded domains consider
x
1/
x
on (0, 1) or (0, 1], and for
bounded ranges consider
x
x
on (0, 1) or (0, 1].
(
x
y
)/(
x
y
)
x
y
x
y
37
20, so
x
y
20
x
y
.
38
1/6.
1/20.
1/
x
1/
a
x
a
xa
(
a
)
a
a
a
).
1
/(1
As
a
gets near to 0, an ever smaller
is required, so there is no one
for all
a
.
39 Taking
y
a
gives the neighbourhood definition of continuity.
40 For the suggested
x
and
y
,
x
y
1/
n
, but
x
y
2, so
there is no universal
for
2 or less.
41 For the suggested
x
and
y
,
x
y
1/
n
, but
f
(
x
)
0 and
f
(
y
)
1, so
f
(
x
)
f
(
y
)
1
, so there is no universal
for
.
42 Take
/
L
.
43
(i) Noneat all.
(ii)
a
x
b
.
(iii) Yes, by qn 4.46, every bounded sequence has a convergent
subsequence.
(iv) Yes, use qn 3.54(v), the difference rule, and the fact that
(
x
) is a null sequence.
(v) By qn 3.78, the closed interval property.
(vi) Both tend to
f
(
c
).
(vii) For suMciently large
n
y
both
f
(
x
)
f
(
c
)
and
f
(
y
)
f
(
c
)
.
44
f
is continuous by qn 26, and therefore uniformly continuous on
[0, 1] by qn 43.
x
y
x
y
ยท
x
y
x
y
provided either
x
or
y
1. To establish uniform continuity on
R
0
, for a given
, choose the lesser of the
from [0, 1] and
.
45
Define
g
(
a
)
lim
f
(
x
) and
g
(
b
)
lim
f
(
x
) and then
g
is continuous
on [
a
,
b
] by qn 6.92. So
g
is uniformly continuous on [
a
,
b
]byqn
43 and therefore uniformly continuous on the subset (
a
,
b
) where
g
coincides with
f
.
