hi. i am having trouble forming a logical chain of reasoning to prove this fact. intuitively i can see it true but i want to prove it rigorously.
from the hypothesis i know since f is integrable that for all e there exists a d such that |Rn - V| < e when |P| < d where Rn is the partial sum of a Riemann sum, V is the limit of these partial sums, and P is a partition of the interval [a,b]. i am trying to prove that there exists some c in the interval [a,b] such that integral (from a to c) f dx = 1/2 integral (from a to b) f dx.
i tried saying that since f is integrable, so is f/2 and it follows that the limit of the Riemann sums of f/2 approaches V/2. now i try to show that there exists a c such that the riemann sums 1/2 (x_i+1 - x_i)f(xi*) from (a to b) = (x_i+1 - x_i) f(xi*) (from a to c) but i am stuck here and can't seem to convince myself of this step.
is this the right way to prove this statement, and if it's not can someone give me some pointers in the right direction? thanks.