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Thread: 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 $\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 $\displaystyle \left| {f(x)} \right| = f^ + (x) + f^ - (x)$
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