# Thread: To prove a function is integrable (or not)

1. ## To prove a function is integrable (or not)

Let $g:[0,2]\mapsto\mathbb{R}$ be defined by $g(x)=\left\{\begin{array}{cc}x,&\mbox{ if }x\epsilon\mathbb{Q}\\-x, & \mbox{ if } x\epsilon\mathbb{R}/\mathbb{Q}\end{array}\right.$

(a) Let P = {0,1,2}. Evaluate U(P,g).
(b) Let P = {0,1,2}. Evaluate L(P,g).
(c) Define a partition Q by $Q={x_0,x_1,...x_n}$ where $0=x_0. Prove that $U(Q,g) \geq 2$.
(d) Is the function g integrable on the interval [0,2]? If so, evaluate $\int^{2}_{0}g(x)dx$. If not, explain why it is not integrable on [0,2].

My works so far:

(a) 3
(b) -3
(c) ....still working
(d) ....still working

Sorry about the amount of problems that I have posted up lately, it is just that there are a lot of new materials that I don't understand from recent lectures (we are working on integrations now), and it is just impossible for me to go to the professor for every single problem.

Thank you very much!

KK

2. The theorem says that a function defined on a closed interval is Riemann integratble if and only if it is almost eveywhere continous. But maybe you did not do that theorem and are asked to show that a bounded ,defined almost discontinous function is not Riemann integrable. If thus, here is something that might help http://www.mathhelpforum.com/math-he...stion-4-a.html. The trick here was to subdivide the the region into rational and then irrational points.

3. So would it be proper to say that this function is not integrable since U(P,g) doesn't equal to L(P,g) for any partition P in g?