Hey everyone! Any input would be appreciated.

A note on notation, on means that is Riemann-Stieltjes integrable with respect to on the interval

Question 1: Let and define on by . Also, let be continuous at . Is on ? If so what is its value.

Answer: Using the following lemma:

Lemma: Suppose that is bounded on , has a finite number of discontinuities and is continuous at all those points, then on .

We can see that since has only one discontinuity, at , and is continuous there.

Next to find the value of this integral consider for any partitions. Now consider the interval created by . Since is a finite point set, it follows that has infinite points. Which indicates it contains values other than . So for any interval on we can see that . Now since we can see that for any partition . Now since it follows that

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Question 2: Let , this is known as Dirchlet's function. Show that for any

Answer: This one is simple I think. Consider any partition of . Now consider any interval formed by . For the same reason as above is infinite. Now since is dense in any subset of the reals it follows that there are value of in such that where it follows similarly to above that for any partition. Now following a similar argument noting that is dense in any subset of the reals we can see that for any partition . So for any partition . Where it follows that since it fails the criterion that given any we must have a partition such that

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Question 3: Does ?

Answer: No. Counterexample . The reason is that which is continuous thus integrable, but for the same reason as in question 2 is not integrable.

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Question 4: Suppose that , is continuous on and . Prove that .

Answer: It is obvious that if . So instead let us prove that if .

Since we can find a Partition of such that for some interval we have that , now since this interval is closed it follows that . Because of this . Now consider that since it follows that .

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Question 5: Prove that if on and that

Answer: Let be partitions of respectively. Let .

Let us note that . To prove this simply note that given an interval and that .

So since taking the infimum over both intervals gives .

Using the same argument except considering that we arrive at

Combining completes the proof.

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Question six: Prove Holder's inequality for integrals, i.e. if then

I can prove this, but it requires me first to prove that if the conditions of are met as above then and I am not sure how to do this.

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For 1-5 any criticism is appreciated ...for number six any help with the inequality would be great.

Thanks in advance