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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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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.

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(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 ...

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