Question

provided integrals on the right exist. If g is a non-increasing function, we have the | again provided the integrals on the right exist. Vte(a,b], l(t)< h(t) → l(t) do(t)2/ h(t) dg(t), a. Throghout these noteill we h oloring dofinition Definition 1.3. Let a, b E R with a < b and let k Zco. We denote by Ck (a, b) the set of functions f such that the k-th derivative f(k) exists and s continuous on an open set U-Uf which satisfies la, b UCR. In particular, Co (fa, b]) denotes the set of functions that are continuous on some open set containing [a, b. We define C (a, b]) by Note that by this definition, we have Co(a, (a, (a, b (a, b). Example 1.4. Assume that g E C(la,b). Then we have f(t) dg(t)-f(t)g'(t) dt. Example 1.5. More generally, assume that g E C(a, b]) and is continuously differentiable on (a-, b+e)-s for some e 0, where S t. tk) la, b] is a finite subset. Assume further that the left and right derivatives of g exist at each t, E S, so that g'(t) in the previous example, we have has at most a jump discontinuity at each ti S. Then as f(t) dg(t)-f(t)g'(t) dt

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?

Answer 1

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,