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This question gives an answer to the equivalence of the two integrals on the space of bounded functions over some interval.

Now suppose the requirement that $f$ be bounded is dropped. Obviously this requires that we define an improper integral in some way, but in general I’m wondering if the properties of each integral are the same on this larger space of functions? For example, are there functions which have an improper Riemann integral over some $[a,b]$ where the corresponding Darboux integral does not exist?

EE18
  • 1,211
  • you can actually show that if $f$ is Riemann-integrable (the actual RIemann definition), then it is necessarily bounded on $[a,b]$. Anyway, for unbounded functions, it is next to impossible to use the Darboux definition, because it is possible to have a function $f$ which is unbounded in both directions in every open interval. So, the lower and upper sums themselves aren’t defined. – peek-a-boo Dec 28 '22 at 23:12
  • If the function is unbounded (positive), the upper Darboux sum will be infinite. – herb steinberg Dec 28 '22 at 23:13

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