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We say that a function $f:[a,b] \to \Bbb R$, ($f$ bounded) is integrable by Darboux if the supremum of the set of the lower Darboux sums is equal to the infimum of the set of the upper Darboux sums or:

$$\sup\{\underline{S}(f,P):P \text{ partition of [a,b]}\}=\inf\{\overline{S}(f,P):P \text{ partition of [a,b]}\}$$

  1. An element of the set differ only by the partition of an interval, each element of the is a Darboux sum with a more refined partition of an interval?

  2. If there are $x\in \{\underline{S}(f,P)\}$ and $y\in\{\overline{S}(f,P)\}$ such that $x=y$, then $f$ must be a constant?

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An answer to $2.$:

If there are such $x$ and $y$ for the same partition i.e.: let $P=\{t_0,...,t_n\}$, be a partition of $[a,b]$ and $x \in \{\underline{S}(f,P)\},y \in \{\overline{S}(f,P)\}$ then it means that $f$ is constant on each interval.

If $x,y$ are such $\forall P$ partitions of $[a,b]$ then it means that $f$ is constant on $[a,b]$

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