Note we are assuming is Riemann integrable. This means that given we can find an such that if is a tagged partition of and , then

Recall that a tagged partition consists of a partition of , represented by a finite sequence of points

,

and a sequence of representatives of the intervals defined by this partition, i.e., a collection of points for

We denote by the number and by the norm of ,

The Riemann sum associated to and is the sum

Pick any partition of with . We want to define a particular tagged partition with underlying partition by appealing to the fact that has an antiderivative . Specifically, by the mean value theorem, we have that for all , there is some such that

Define in terms of the points and the partition . Then

By our choice of , we know that , so

But the left hand side is independent of , and was arbitrary. It follows that , as we wanted.

The same argument, but restricting ourselves to the interval , shows that

,

where for , so in particular . It follows that and differ by a constant, and therefore, if is Riemann integrable and has an antiderivative at all, then is such an antiderivative.

The question remains of what Riemann integrable functions (with the intermediate value property) have antiderivatives. Another natural question has to do with the fact that our definition of antiderivative is very restrictive; it also makes sense to simply ask whether the equality must hold for some , assuming only that is integrable. It turns out that both questions require the introduction of the Lebesgue integral to be answered in a satisfactory way.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Monday, February 6th, 2012 at 12:50 pm and is filed under 515: Analysis II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

No, the rank of a set $x$ is the least $\alpha$ such that $x\in V_{\alpha+1}$. Note that if $\alpha$ is limit, any $x\in V_\alpha$ belongs to some $V_\beta$ with $\beta

The real numbers are the usual thing. Surreal numbers are not real numbers, so no, they are not an example of non-constructible reals. Any real $r$ can be written as an infinite sequence $(n;d_1,d_2,\dots)$ where $n$ in an integer and the $d_i$ are digits. Whether the real is rational, constructible or not, is irrelevant. Any rational number, in fact, any al […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

This is a beautiful and truly fundamental result, and so there are several good quality presentations. Try MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3, or any of the newer editions (the 2003 second ed […]