inverse functions
increasing function
given , a function
f
:
A
→
R
is increasing if given
x
,
y
∈
A
with
x
<
y
we have
f
(
x
)
≤
f
(
y
)
. An increasing function is also known as a non-decreasing function.
strictly increasing function
given
A
⊆
R
, a function
f
:
A
→
R
is strictly increasing if given
x
,
y
∈
A
with
x
<
y
we have
f
(
x
)
<
f
(
y
)
.
decreasing function
given
A
⊆
R
, a function
f
:
A
→
R
is decreasing if given
x
,
y
∈
A
with
x
<
y
we have
f
(
x
)
≥
f
(
y
)
. A decreasing function is also known as a non-increasing function.
strictly decreasing function
given
A
⊆
R
, a function
f
:
A
→
R
is decreasing if given
x
,
y
∈
A
with
x
<
y
we have
f
(
x
)
>
f
(
y
)
monotone function
a function defined on a subset of
R
is said to be monotone if and only if it is non-increasing or non-decreasing
strictly monotone function
a function defined on a subset of
R
is said to be monotone if and only if it is stictly increasing or strictly decreasing
invertible function
Let
f
:
X
→
Y
be a function, if there is a function
g
:
Y
→
X
such that
g
(
f
(
x
)
)
=
x
for all
x
∈
X
and
f
(
g
(
y
)
)
=
y
for all
y
∈
Y
a function is invertible iff it is bijective
TODO
If
f
is strictly monotone then it is invertible
TODO
Suppose that
f
:
X
→
Y
is strictly monotone, then
f
−
1
:
Y
→
X
is a continuous function
Suppose without loss of generality that
f
is strictly increasing on
I
=
(
a
,
b
)
.
We'll show that the function is continuous, so let
p
∈
I
and we'll show that
lim
y
→
p
f
−
1
(
x
)
=
f
−
1
(
p
)
using the epsilon delta definition.
Let
ϵ
∈
R
>
0
, let
x
=
f
−
1
(
p
)
and note that
x
−
ϵ
<
x
+
ϵ
and therefore since
f
is strictly increasing we know that
f
(
x
−
ϵ
)
<
f
(
x
+
ϵ
)
, in other words, there is some
δ
1
,
δ
2
∈
R
>
0
such that
f
(
x
−
ϵ
)
=
p
−
δ
1
and
f
(
x
+
ϵ
)
=
p
+
δ
2
, now take
δ
=
min
(
δ
1
,
δ
2
)
and suppose
|
y
−
p
|
<
δ
.
if
|
y
−
p
|
<
δ
, then
δ
−
p
<
y
<
δ
+
p