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Math Help - Riemann integrable, function

  1. #1
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    Riemann integrable, function

    Suppose that f:[a,b] \rightarrow \mathbb{R} is a Riemann integrable on [a,b] and f(x) \geq 0 for all  x \in [a,b]. Prove that \sqrt{f} is Riemann integrable on [a,b].
    Relevant Theorems & Definitions
    Definition - Riemann integrable - if upper integral of f(x)dx= lower integral of f(x)dx.
    Theorem - Riemann integrable iff \exists A \in \mathbb{R} such that \forall \epsilon >0 \exists a partition P of [a,b] such that \forall marked refinements Q of P, |S(Q, f) - A | \leq \epsilon, where A=\int^a_b f(x)dx.
    Theorem - Let f: [a,b] \rightarrow \mathbb{R} be a continuous function on [a,b]. Then f is Riemann integrable on [a,b].
    etc.
    Theorem - Riemann integrable iff for each \epsilon > 0 \exists a partition P of [a,b] such that U(P,f)-L(P,f) \leq \epsilon.

    I haven't gotten very far doing this problem. I need help with this problem. Thank you.
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  2. #2
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    This can be more "easily" solved if you work with the following.
    Suppose f is Riemann integrable, and g continuous, on [a,b]. Then g\circ f is Riemann integrable on [a,b].

    Hint to proof: First consider a partition P such that U(P.f)-L(P.f)<\delta^2. Then make two sets as follows; A:=\{i:M_i-m_i<\delta\} and B:=A^c. You choose \delta needed for the definition of continuity for g. Also since f is Riemann integrable, it is bounded on this interval (you will need this).
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