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If $R(f,P,T) <0$ for every partition pair P,T on the interval $[a,b]$, then $\int_{a}^b f(x) dx<0.$

I think this is false since the Riemann sum may exist for $f(x)$; however, it does necessarily mean that the integral exists.

Similarly, If $R(f,P,T) =0$ for every partition pair P,T on the interval $[a,b]$, then $\int_{a}^b f(x) dx=0.$

I think this is false for similar reasons.

Or am I getting this wrong?

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I take it $T$ is a set of tags for the partition $P$? –  David Mitra Jan 23 '13 at 1:29
    
For your second problem, note for any $x\in[a,b]$, by considering $P=\{[a,b]\}$ and $T=\{x\}$, one deduces $f(x)=0$. –  David Mitra Jan 23 '13 at 1:30
    
Yes. I should have clarified that. –  user43901 Jan 23 '13 at 1:31
    
@DavidMitra: would you be willing to walk to me through an intuitive explanation of the problem? I am finding hard to build intuition in these problems. –  user43901 Jan 23 '13 at 1:32
    
My issue is I can have a $f(x)=x$ from $-1$ to $1$, then the area under the curve is zero, but $f(x)$ is not; how did you deduce that $f(x)=0$? –  user43901 Jan 23 '13 at 1:38
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