Graphics Reference
In-Depth Information
Proof.
Easy.
Recall that every continuous function assumes its minimum and maximum value
on a closed interval, so that the next definition is well-defined.
Definition.
Let f :
R
Æ
R
be a continuous function. The function
()
Æ
()
fI
I
:
RR
I
defined by
()
=
()
()
Œ
()
f
A
[min
f a
, max
f a
],
for A
I
R
,
I
aA
Œ
aA
Œ
is called the
induced function
.
18.2.8
Theorem.
The induced function of a continuous function is a continuous
function on I(
R
).
Proof.
This is easy to prove from the definitions.
With this notation of induced maps we shall free to write expressions such as
kA
A
,
e
,ln
A
,sin
A etc
,
.
Furthermore, from Theorem 18.2.8 all of these are continuous.
18.2.9
Example.
If f(x) = 2x + 3, then f
I
([a,b]) = [2a + 3,2b + 3].
If f(x) = x
2
, then
18.2.10
Example.
)
=
[
]
(
[
]
22
fab
,
ab
,
,
fa
≥
0
,
I
=
[
]
22
b
,
a
,
if b
£
0
,
[
(
)
]
22
=
0
,max
a
,
b
,
otherwise
.
Definition.
Define the
absolute value
of an interval A = [a,b] in I(
R
), denoted
by |A|, by
(
[
]
)
=
(
)
=
{
}
AdA
=
,
00
,
max
ab
,
max
xxA
Œ
.
18.2.11
Example.
|[-1,3]| = 3.
18.2.12 Lemma.
Let A, B, C ΠI(
R
) and x Œ
R
. The absolute value function for
intervals has the following properties:
(1) |A| ≥ 0 and |A| = 0 if and only if A = [0,0].
(2) If A Õ B, then |A| £ |B|.
(3) |A + B| £ |A| + |B|.
