In addition to this, I want to show that if converges, then we have
I agree, integration by parts is definitely the right way to go about this. In fact, . When you divide by , the term in square brackets will go to 0 as . So you just need to estimate the size of the integral on the right.
You know that F(t) is bounded, say for some constant M. Then . Again, when you divide by , that expression will go to 0 as .
This is a bit more subtle than the case where p>1. But you have the additional information that exists as the limit . So F is continuous on the left at b, and given there exists a point such that whenever x and t lie in the interval .
To estimate , split it up as . The first of those two integrals is independent of x, so when we divide by g(x) and let , it will go to 0. Thus we need only look at the second of the two integrals, .
Integrating by parts as previously, we get . Use the fact that to write this as . The first two terms on the right side of that equation are bounded, so when we divide by g(x) and let they will go to 0. Thus we need only estimate the size of the integral term.
For that, using the inequality , we get . When we divide by g(x), that gives a result that is essentially less than .
Putting everything together, that should show that .
I don't think it's as simple as that, because you don't know that is bounded. In fact, as , so that integral might well be unbounded.
It's a constant as far as t is concerned, and the integral is with respect to t. After all, x is the upper limit of integration in the integral , so x has to be regarded as a constant for the purposes of that integral.
The reason that F is continuous at b comes from the definition of an improper integral. To say that the integral converges means that , which is the same as saying that F is continuous at b.