Suppose that and has an antiderivative . Then
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.