In the definition of Darboux integral of a bounded function $f:[a, b]\rightarrow \mathbb{R}$, partitions of $[a, b]$ is not restricted to equidistant.
It can be proved that if the upper Darboux sum and lower Darboux sum converges to the same value for equidistant partition of the interval, $f$ is integrable.
Is there a counterexample such that $f$ is Darboux integrable but its equidisant upper Darboux sum and lower Darboux sum don’t converge to same value?