# partitions of function

• Apr 10th 2010, 04:01 AM
charikaar
partitions of function
Define f:[-2,2]---->R by
$f(x)=x^{2}$, x belongs to irrationals.
$f(x)=0$ 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 $U(f,P)\geq1$
do we just say f has supremum greater than 1 in any subinterval. Therefore $U(f,P)\geq1$ for any partion P.
iii) does $\int_{-2}^2f(x)dx$ exist?

Thank for any help.
• Apr 10th 2010, 07:40 AM
Cut the integral into $\int_{-2}^2 = \int_{-2}^{-1}+\int_{-1}^0 + \int_0^1 + \int_1^2$. On [-2,-1] and [1,2], every partition can be chosen so that the sampled value is $\ge 1$ so the upper sum is greater than 2.