**Remark**: We know that piecewise monotonic continuous functions on
are
nice. It turns out that every continuous function on is nice, but we
are not
going to prove this. The theorems stated in this chapter for nice functions
are usually stated for continuous functions. You can find a proof that every
continuous
function on an interval is integrable on (and hence that every
continuous function on is nice on ) in [44, page 246]
or in
[1, page 153]. However both of these sources use a slightly
different
definition of continuity and of integral than we do, so you will need to do
some work
to translate the proofs in these references into proofs in our terms. You
might try
to prove the result yourself, but the proof is rather tricky. For all the
applications we will make in this course, the functions examined will be
continuous
and piecewise monotonic so the theorems as we prove them are good enough.

Proof: By the definition of antiderivative, is continuous on and on . Let be arbitrary points in . I will suppose . (Note that if (16.4) holds when , then it holds when , since both sides of the equation change sign when and are interchanged. Also note that the theorem clearly holds for .)

Let
be any partition of , and let be an
integer
with . If we can apply the mean value theorem to
on
to find a number
such that

If , let . Then is a sample for such that

We have shown that for every partition of there is a sample for such that

Let be a sequence of partitions for such that , and for each let be a sample for such that

Then, since is integrable on ,

and

have the same area - a rather remarkable result.

Proof: Since is continuous on we can find numbers
such
that

By the inequality theorem for integrals

(here and denote constant functions) i.e.,

i.e.,

We can now apply the intermediate value property to on the interval whose endpoints are and to get a number between and such that

The number is in the interval , so we are done.

Proof: Let . I will show that is continuous at .
Since is integrable on there is a number such that

By the corollary to the inequality theorem for integrals (8.17), it follows that

for all . Thus, for all ,

Now , so by the squeezing rule for limits of functions,

. It follows that .

Proof: Let

and let be a point in . Let be any sequence in such that . Then

By the mean value theorem for integrals, there is a number between and such that

Now

and since , we have , by the squeezing rule for sequences. Since is continuous, we conclude that ; i.e.,

i.e.,

This proves that for . In addition is continuous on by lemma 16.9. Hence is an antiderivative for on .

**Remark** Leibnitz's
statement of *the fundamental principle of the calculus* was
the following:

Differences and sums are the inverses of one another, that is to say, the sum of the differences of a series is a term of the series, and the difference of the sums of a series is a term of the series; and I enunciate the former thus, , and the latter thus, [34, page 142].To see the relation between Leibnitz's formulas and ours, in the equation , write to get , or . This corresponds to equation (16.11). Equation (16.4) can be written as

If we cancel the 's (in the next chapter we will show that this is actually justified!) we get . This is not quite the same as . However if you choose the origin of coordinates to be , then the two formulas coincide.

To emphasize the inverse-like relation between differentiation and
integration, I will restate our formulas for both parts of the
the fundamental theorem, ignoring all hypotheses:

By exploiting the ambiguous notation for indefinite integrals, we
can get a form almost identical with Leibniz's:

We will calculate the derivatives of , and . By the fundamental theorem,

Now , so by the chain rule,

We have , so by the product rule,

**a)****b)****c)****d)****e)**- .

- a)
- .
- b)
- .
- c)
- .