Let $P = \{p_i\}_{i= 1, n}$, $P' = \{p'_j\}_{j = 1, m}$ be partitions of an interval with max$|p'_j| \le $min$|p_i|$, i.e. all the sub-intervals of $P'$ are at least as short as all the sub-intervals of $P$.
Let $f$ be a bounded function on the interval.
Let $U_{P, f}, L_{P, f}$ be the upper and lower Darboux sums of $f$ on $P$ (and correspondingly on $P'$).

There is a proof here https://math.stackexchange.com/q/353810 in the lemma that $U_{P', f} - L_{P', f} \le 3(U_{P, f} - L_{P, f} )$.

I assume from this that it is not generally the case that $U_{P', f} - L_{P', f} \le U_{P, f} - L_{P, f}$ and I'm looking for a counter-example to show this is so. I do know that this tighter inequality holds when $P'$ is a refinement of $P$ but even so I cannot construct a counter-example.

After further thought .....

Let $P = \{0, 1/2 , 1\} $ and $P' = \{0, 1/3, 2/3, 1\} $ be partitions of $[0, 1]$. Define $ f $ on $[0, 1]$ by $f(7/12) = 1; f(1) = 1; f(x) = 0 $ otherwise.

Then $U_{P', f} - L_{P', f} = 0 + 1/3 + 1/3 = 2/3 > 1/ 2 = U_{P, f} - L_{P, f} $

  • $\begingroup$ If you have answered your own question, shouldn't you post the answer as an answer, instead of appending it to the question? $\endgroup$ – bof Feb 20 at 9:43
  • $\begingroup$ @bof I have once or twice in the past, but now wonder if it doesn't just look like going for the points ? $\endgroup$ – Tom Collinge Feb 20 at 10:57

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