If |f| is Riemann integrable, then f is Riemann integrable???
So i am stuck here.. how do i prove the first & second inequalities?
Also if |f| is Riemann integrable, then f is Riemann integrable.
I think it's ture but i dont know how to prove it.
any hints would be appreciated! Thank You
Solution 1:
Answer is No.
Counter Example:
Define $f:[0, 1]\rightarrow \mathbb R$ as follows $f(x)= 1$ if $x\in (\mathbb R - \mathbb Q ) \cap[0,1]$; $f(x)= - 1$ if $x\in \mathbb Q \cap [0,1]$. Then $|f|\in R[0,1]$(as being a continuous function) but $f \not \in R[0,1]$ (choosing partition one can conclude by definition).