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Math Help - Riemann Integrability absolute value

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    Riemann Integrability absolute value

    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))
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    Quote Originally Posted by ns131005 View Post
    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 f^ +  (x) = \left\{ {\begin{array}{*{20}c}<br />
   {f(x),} & {x \geqslant 0}  \\<br />
   {0,} & {x < 0}  \\<br /> <br />
 \end{array} } \right.\,\& \,f^ -  (x) = \left\{ {\begin{array}{*{20}c}<br />
   { - f(x),} & {x \leqslant 0}  \\<br />
   {0,} & {x > 0}  \\<br /> <br />
 \end{array} } \right.
    Show that each of those functions is Riemann integrable.
    Then note that \left| {f(x)} \right| = f^ +  (x) + f^ -  (x)
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