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    Proofs : Riemann Integrable Functions

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    Let RI be the set of functions that are Riemann Integrable.

    Disprove with a counterexample or prove the following true.

    (a) f in RI implies |f| in RI

    (b) |f| in RI implies f in RI

    (c) f in RI and 0 < c <= |f(x)| forall x implies 1/f in RI

    (d) f in RI implies f^2 in RI

    (e) f^2 in RI implies f in RI

    (f) f^3 in RI implies f in RI

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    Solution Preview

    (a) True
    Proof: Let and . Then . Since
    is Riemann integrable, then both and are Riemann integrable. Thus is also Riemann integrable. Hence belongs to RI.
    (b) False
    For example, we consider defined as if ...

    Solution Summary

    Riemann integrability is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.