Definition. A function is called
**one-to-one** if
whenever

Definition. For any function
,
the **inverse** of
,
denoted by
,
is the set of all pairs
for which
is in

Theorem. is a function if and only if is one-to-one.

Proof. (1) Assume is one-to-one. Let , be two pairs in . Then and are in . This means . Since is one-to-one, it follows that . Hence is a function.

(2) Assume that is a function. If , then contains the pair and . Therefore and are in . Since is a function, this implies . Hence is one-to-one.

Theorem. If
is continuous and one-to-one on an interval, then
is either **increasing** or **decreasing** on that
interval.

Proof. If
is neither increasing nor decreasing, then there exist
and
on the interval with
and
.
Consider the function
defined by

Then
is continuous on
with
and
.
It follows from the *Intermediate Value Theorem* that there exists
such that
.
This implies
since
is one-to-one. But the inequality

shows that
,
a contradiction.

Theorem. If is continuous and one-to-one on an interval, then is also continuous.

Proof. From the preceding theorem, we see that
is either increasing or decreasing on that interval. By replacing
,
we may assume that
is increasing. Fix
in the domain of
and let
.
Such a number
must be of the form
for some
in the domain of
.
We wish to find
so that the inequality

implies the inequality

Now let
be the smaller of
and
.
This choice of
ensures that

Consequently, the inequality

implies the inequality

Since
is also increasing, we have

i.e.,

Theorem. If
is a continuous one-to-one function defined on an interval and
then
is **not** differentiable at
.

Proof. Since
,
the differentiability of
at
would imply

a contradiction.

Theorem. Let
a continuous one-to-one function defined on an interval, and suppose that
is differentiable at
,
with derivative
.
Then
is differentiable at
,
and

Proof. Let
.
Every number
in the domain of
can be written in the form

for some unique
.
From this we have

or

The previous theorem shows that
is continuous at
.
Therefore

i.e.,

Since

it follows from

that

Definition.

Definition. For
,
define

It follows from the Fundamental Theorem of Calculus that
is differentiable on
and

Therefore decreases from to

Definition. If
,
then
is the unique number in
such that

Definition.

Theorem. If
,
then

Proof. Let
.
Then
.
Since
is differentiable with
,
it follows that
is differentiable with
.
Hence

Therefore
.
Since
we have

For
,
define

For any real number
,
write

where
and an integer so that
.
Define

Then

Other standard trigonometric functions are defined as:

Theorem. If
then

If
,
then

The inverse of the function

is denoted by
or arcsin. Thus
is the unique number in
satisfying

The domain of
is
.
The inverse of

is denoted by
or arccos. The domain of
is
The
is the unique number in
satisfying

The inverse of the function

is denoted by
or arctan. The domain of
is
.
The
is the unique number in
satisfying

Theorem. For
,
we have

For
,
we have

Theorem. The only solution to the differential equation

is

Proof. The given condition implies

Therefore

so
is constant. From
it follows that this constant is zero. Thus

This implies

Theorem. If
has second derivative everywhere and

then

Proof. Let

Then

Consequently,

Therefore

Theorem.

Proof. Fix
.
Define
.
Then

Therefore

The
previous theorem shows