Problem Statement: Let be defined on . (a) Find . (b) State a definition of Riemann Integrable and use it to prove or disprove is Riemann Integrable on .
Note that for and that for . We know that if and only if for some . With this in mind we may re-write the above limit as . If then it follows that for some and so we may write .
So, if it follows that and thus, . If then for some values of but as all of these functions go to . This gives us that if and if .
(b) Claim: is not Riemann Integrable on . A function is Riemann Integrable on [a,b] if for every there exists a such that if and are partitions of [a,b] such that and then it follows that . Note that is the associated point in and that is the associated point in .
Assume that is Riemann Integrable on [a,b]. Let , then there exists a such that . Since this must be true for any associated points in and we may choose our associated points.
Pick associated points for each and for each . We may do this since is dense in . Since is Riemann Integrable this implies that
We know that since and so the above statement is a contradiction. Thus, our assumption that is Riemann Integrable must be incorrect.
Reflection: Wow, what a problem. The qual writers had a lot of fun writing this one. They gave us this horribly, nasty looking thing that reduced down to the Dirichlet Function. Wow. Ok, enough about the problem itself, now to talk about the math. The hard part of this problem was realizing that was the Dirichlet function. Once we got to there showing it was not Riemann Integrable was a piece of cake. Finding really rested on the fact that we could rewrite . Since we can bound between and we know that it will go to if , otherwise, will always equal and so it’s limit will also be . This trick with this problem is to recognize which values of cause the function to go to and which values cause the function to go to . After recognizing that this depends on if is an integer or not we are able to get the Dirichlet Function and the rest is a piece of cake.