When we say that a function is Riemann or Darboux integrable, do the partitions matter? In other words, suppose that we have a closed interval $[a,b]$ and a function $f$ that is bounded on said interval. If one can find only one partition such that $U(f)=L(f)$, would that be enough to show that the function $f$ is integrable on $[a,b]$? Or does the equality have to hold regardless of the choice of partition?
(Note: here $U(f)$ is the upper Darboux integral and $L(f)$ is the lower Darboux integral).