How would I prove 1.4. What is this called? I can't find these properties in my textbook. What is the name of this stuff?
Let . Since is integrable with respect to , that is,
exists, there is some such that for all . On the other hand, since is continuous (because ) on the compact interval , it is uniformly continuous; there is such that for all , if then
Let be a partition of such that for each . Since , the derivative is continuous; therefore, on any subinterval , there is some such that
we use mean value theorem on to obtain
for some .
Then the upper Riemann sum satisfies
On the other hand,
Therefore,
Since , this shows that
Therefore,
How would I prove 1.4. What is this called? I can't find these properties in my...
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