I am trying to prove that if f is remann integrable then so is absolute value of f. I am supposed to use remann's criterion. I think the triangle inequality is going to factor in but I am not sure how to relate U(P,f) with U(P,abs(f))
I am trying to prove that if f is remann integrable then so is absolute value of f. I am supposed to use remann's criterion. I think the triangle inequality is going to factor in but I am not sure how to relate U(P,f) with U(P,abs(f))
There are as many proofs for this as there are textbooks.
Define $\displaystyle f^ + (x) = \left\{ {\begin{array}{*{20}c}
{f(x),} & {x \geqslant 0} \\
{0,} & {x < 0} \\
\end{array} } \right.\,\& \,f^ - (x) = \left\{ {\begin{array}{*{20}c}
{ - f(x),} & {x \leqslant 0} \\
{0,} & {x > 0} \\
\end{array} } \right.$
Show that each of those functions is Riemann integrable.
Then note that $\displaystyle \left| {f(x)} \right| = f^ + (x) + f^ - (x)$