Theorem 2 (6.24, Intermediate Value Theorem)
Suppose
f
: [
a, b
]
→
R
is continuous, and
f
(
a
)
< d < f
(
b
)
. Then there
exists
c
∈
(
a, b
)
such that
f
(
c
) =
d
.
Proof:
We did a hands-on proof already. Now, we can simplify
it a bit. Let
B
=
{
t
∈
[
a, b
] :
f
(
t
)
< d
}
a
∈
B
, so
B
=
∅
. By the Supremum Property, sup
B
exists and
is real so let
c
= sup
B
. Since
a
∈
B
,
c
≥
a
.
B
⊆
[
a, b
], so
c
≤
b
.
Therefore,
c
∈
[
a, b
]. We claim that
f
(
c
) =
d
.
Let
t
n
= min
⎧
⎪
⎨
⎪
⎩
c
+
1
n
, b
⎫
⎪
⎬
⎪
⎭
Either
t
n
> c
, in which case
t
n
∈
B
, or
t
n
=
c
=
b
, in which case
f
(
t
n
)
> d
, so again
t
n
∈
B
; in either case,
f
(
t
n
)
≥
d
. Since
f
is
continuous at
c
,
f
(
c
) = lim
n
→∞
f
(
t
n
)
≥
d
(Theorem 3.5 in de la
Fuente).
Since
c
= sup
B
, we may find
s
n
∈
B
such that
c
≥
s
n
≥
c
−
1
n
Since
s
n
∈
B
,
f
(
s
n
)
< d
.
Since
f
is continuous at
c
,
f
(
c
) =
lim
n
→∞
f
(
s
n
)
≤
d
(Theorem 3.5 in de la Fuente).
Since
d
≤
f
(
c
)
≤
d
,
f
(
c
) =
d
. Since
f
(
a
)
< d
and
f
(
b
)
> d
,
a
=
c
=
b
, so
c
∈
(
a, b
).
Monotonic Functions:
Definition 3
A function
f
is
monotonically increasing
if
y > x
⇒
f
(
y
)
≥
f
(
x
)
2