
partitions of function
Define f:[2,2]>R by
$\displaystyle f(x)=x^{2}$, x belongs to irrationals.
$\displaystyle f(x)=0$ otherwise.
Suppose P is a partition of [1,2].
i) Show that L(f,P)=0
f has infimum zero in any subinteral. Therefore L(f,P)=0 for any partition P.
ii) How do i show $\displaystyle U(f,P)\geq1$
do we just say f has supremum greater than 1 in any subinterval. Therefore $\displaystyle U(f,P)\geq1$ for any partion P.
iii) does $\displaystyle \int_{2}^2f(x)dx$ exist?
Thank for any help.

Cut the integral into $\displaystyle \int_{2}^2 = \int_{2}^{1}+\int_{1}^0 + \int_0^1 + \int_1^2$. On [2,1] and [1,2], every partition can be chosen so that the sampled value is $\displaystyle \ge 1$ so the upper sum is greater than 2.
The Riemann integral exists iff the lower and upper Riemann sums tend to the same limit as the partition refines. Since the lower one is always zero, and the upper one is greater than 2, they do not.