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If there exists partition $P$ in interval $[a, b]$, $P = {x_0, x_1, ... x_n}$ s. t. $U(f, P) > 0$, then the upper integral is $>0$

It's part of the three true or false questions, I've done the other 2 but here I can't think of a counterexample, and if it's true I do not know how to prove it.

Also $f: [a, b] \to \mathbb R$ is Riemann integratable function

repulsive23
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  • What about a function which is $1$ on the integers and zero elsewhere, for $P$, you choose a partition and points to evaluate that are all integers? – Michael Burr Feb 08 '17 at 17:32
  • Wait, I dont even have to go that far, I can just make a function that is $1$ on $a$ and $0 \in (a,b]$? and with a partition $P = a, b$ this works, since the upper integral will be zero, right? – repulsive23 Feb 08 '17 at 17:45
  • In principle, yes. There are a few different definitions of these concepts (that are more or less equivalent), so it depends on the definition that you're using, but this is the right idea. – Michael Burr Feb 08 '17 at 18:00

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