In a blog titled “Introducing Lady L“, we showed that

In light of the fact that is a monotonic function on , i.e.,

,

we can prove that

Let is *continuous* and *monotonic*

The proof is simple, rigorous and similar to what we have done in “Introducing Lady L“.

Let be a monotonically increasing function,

.

and

.

Since

It follows that

.

The fact that is continuous tells us

.

In addition,

.

Therefore,

.

The case for can be handled in a similar fashion.

(2) becomes more general when the condition of being a monotonic function is removed:

Let is *continuous*

Let’s prove it.

By definition, is continuous at means

.

It implies .

For , we have

That is

Since ,

it follows that

or,

By (3), we have

i.e.,

As a result,

.

For , since ,

.

That is

or,

Divide throughout, and express as , we arrived at

as before.

We are now poised to define the derivative of a function:

Let be a function on an opensubset of . Let . We say that is differentiable at if

exists. If exists, this limit, commonly denoted by or , is called the derivative of at .

For function , the difference of two differentiable functions,

.

By definition,

We have

or,

With this definition, we can also re-state (4) as:

Let is *continuous*

From (7), it is clear that is a solution of the following equation:

where is the unknown function.

In fact, for *any* function that satisfies (8), we have

.

Therefore by (6),

That is, is a function whose derivative is everywhere zero.

Geometrically, if the curve of a function is horizontally directed at every point, it represents a constant function.

It is even more obvious if one considers a function that describes the position (on some axis) of a car at time . Then the derivative of the function, is the instantaneous velocity of the car. If the derivative is zero for some time interval (the car does not move) then the value of the function is constant (the car stays where it is).

Hence, we assert

A function on an open interval has derivative zero at each point , a constant.

From (9) and above assertion, whose rigorous proof we postpone until later in “Sprint to FTC“, it follows that

or,

where c is a constant.

We know

.

It implies

.

i.e.,

and, (10) becomes

Let , we have

This is the well known Newton-Leibniz formula. It expresses an algorithm for evaluating the definite integral :

Find *any* function whose derivative is , and the difference gives the answer.