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Thread: Riemann integrable, function

  1. #1
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    Riemann integrable, function

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

    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 $\displaystyle f$ is Riemann integrable, and $\displaystyle g$ continuous, on $\displaystyle [a,b]$. Then $\displaystyle g\circ f$ is Riemann integrable on $\displaystyle [a,b]$.

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