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,
It follows that
The fact that is continuous tells us
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
it follows that
By (3), we have
As a result,
For , since ,
Divide throughout, and express as , we arrived at
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,
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
where c is a constant.
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.