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 [sci.math 17 Mar 2000 13:47:26 GMT] 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 . 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 . Dave L. Renfro