From: dlrenfro@gateway.net (Dave L. Renfro)
Subject: Re: Why is an integral the antiderivative?
Date: 18 Mar 2000 20:00:22 -0500
Newsgroups: sci.math
Summary: [missing]

Edward C. Hook <hook@nas.nasa.gov>
[sci.math 17 Mar 2000 13:47:26 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/zheldkheetwar>

wrote (in part, beginning with a quote from Michael Press'
       March 16, 2000 post in the same thread)

> |> 
> |> Here is how my "thinking" went.   f attains a maximum (or
> |> minumum). f' changes sign at the extremum.  By the IVT f'
> |> is zero somewhere on (a,b).  I plead nolo contendere.
> |>
> 
>    Probably a wise choice ... :-)
> 
>    That would _almost_ work, except that you'd need to
>    assume that f' is continuous on the interval [a,b].
>    So it doesn't fly ...

Actually, it does fly. Regardless of how badly behaved a
derivative is, every derivative satisfies the Intermediate
Value Property (IVP). You can find this proved in some
undergraduate real analysis texts. [If f'(x1) < d < f'(x2), then
g(x) = f(x) - d*x is differentiable (hence also continuous).
Let x=c be where g attains its minimum value on [x1, x2].
(This uses the continuity of g on [x1, x2].) Then
g'(c) = 0 ==> f'(c) = d. Finally, because g'(x1) < 0 and
g'(x2) > 0, c can't be x1 or x2. Hence, we've shown there
exists c such that x1 < c < x2 and f'(c) = d.] This was first
proved by G. Darboux in 1875.

In fact, derivatives satisfy the Denjoy-Clarkson property,
a property much stronger than the IVP. The IVP implies
that each set E(a,b) = {x: a < f'(x) < b} is either empty
or infinite. (It is not difficult to show that "cardinality
of the continuum" can replace "infinite".) Denjoy (1916) (and
later rediscovered by J. A. Clarkson in 1947) improved this
by showing that each of the sets E(a,b) is either empty or
has positive measure. For an application of the Denjoy-Clarkson
property, see part IV of my sci.math post titled "HISTORICAL ESSAY
ON CONTINUITY OF DERIVATIVES" at

<http://forum.swarthmore.edu/epigone/sci.math/pringquibroi>.

Some remarks about discontinuous functions having the IVP can be
found in my Nov. 25, 1999 alt.math.undergrad post (see my discussion
of Question 2) under the thread titled "real analysis" at

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/kayglanwhel>.

Dave L. Renfro