Hello, I have to prove for a monotonic decreasing function f(x) on , [0,4] prove that the integral from 0 to 4 f(x)dx exists. The proof I have is similar to this:
Proof. If f is constant, then we are done. We prove the case for f monotone increasing. The case for monotone decreasing is similiar. We again use the condition (*) to prove that f is Riemann-integrable. If , we set and consider any partition P with . Since f is monotone increasing on [a,b], then and . Hence
How does this change for a decreasing function? Does the partition in anyway change?