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

(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 $\displaystyle Q={x_0,x_1,...x_n}$ where $\displaystyle 0=x_0<x_1<...<x_{n-1}<x_n=2$. Prove that $\displaystyle U(Q,g) \geq 2$.

(d) Is the function g integrable on the interval [0,2]? If so, evaluate $\displaystyle \int^{2}_{0}g(x)dx$. 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