Graphics Reference
In-Depth Information
(48) For positive
a
1 and positiverational
x
wecan construct
a
by
qn 4.40, and wecan define
a
1 and
a
1/
a
.
(i) Provethat
f
: Q
R
defined by
f
(
x
)
a
is strictly
monotonic. (In fact it is monotonic increasing when 1
a
,
and monotonic decreasing when 0
a
1.)
(ii) For
x
1)/
x
. Useqn 2.50(iii) to
show that
g
(
n
)
g
(
m
) for two positive integers
n
m
.By
putting
b
a
provethat
g
(
p
/
q
)
g
(
r
/
s
) for two rational
numbers 0
p
/
q
r
/
s
, so that
g
: Q
R
is monotonic
increasing. Use qn 4.41 to extend
g
to a monotonic increasing
function on
0, and
a
1, let
g
(
x
)
(
a
.
(iii) Provethat
f
satisfies a Lipschitz condition on any closed
interval [0,
B
]
Q
0
, where
B
is a positiverational and so
f
is
uniformly continuous on that interval by qn 42.
(iv) Deduce from qn 47 that
f
may be extended to a continuous
function on [0,
B
] in a uniqueway.
(v) Usethedefinition of
a
Q
to extend
f
to a continuous function
on [
B
,
B
].
Summary
Definition
qns 1, 2
Monotonic functions
A real function
f
is said to bemonotonic
increasing when
x
y
f
(
x
)
f
(
y
).
A real function
f
is said to bestrictly monotonic
increasing when
x
y
f
(
x
)
f
(
y
).
A real function
f
is said to bemonotonic
decreasing when
x
y
f
(
x
)
f
(
x
)
f
(
y
).
A real function
f
is said to bestrictly monotonic
decreasing when
x
y
f
(
x
)
f
(
y
).
Theorem
qns 5, 6, 7
A monotonic function has one-sided limits at
each point of its domain. It is continuous where
the two one-sided limits are equal.
Definition
qns 8, 9
Intervals
A subset
I
of R is said to be
connected
if when
a
,
b
I
, and
a
b
, then
x
I
for every
x
satisfying
a
x
b
.
A connected set is called an interval.
Bounded intervals are classified as
closed intervals [
a
,
b
], open intervals (
a
,
b
),
half-open intervals (
a
,
b
] and singletons
a
.
