Riemann integrable, function

**diondex223** · April 15th 2009, 06:27 PM

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.

**putnam120** · April 23rd 2009, 08:09 AM

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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