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.

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This is a very interesting question (and I really want to see what other answers you receive). I do not know of any general metatheorems ensuring that what you ask (in particular, about consistency strength) is the case, at least under reasonable conditions. However, arguments establishing the proof theoretic ordinal of a theory $T$ usually entail this. You […]

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There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]

You may be interested in the following paper: Lorenz Halbeisen, and Norbert Hungerbühler. The cardinality of Hamel bases of Banach spaces, East-West Journal of Mathematics, 2, (2000) 153-159. There, Lorenz and Norbert prove a few results about the size of Hamel bases of arbitrary infinite dimensional Banach spaces. In particular, they show: Lemma 3.4. If $K\ […]

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