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Mathematics
Topology and Analysis
Original definition of Riemann Integral and Darboux Sums
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[QUOTE="nuuskur, post: 6851283, member: 519618"] A map defined in a closed interval is Riemann integrable if and only if it is bounded and the subset of its discontinuity points has measure zero. A definition need not apply in any practical sense. If it renders the concept "well defined" that's good enough. We can prove if and only if theorems for the initial definition later. Any of the equivalent conditions can be regarded as definition for the initial concept. It's good to have qualitatively different conditions equivalent to the same thing - in different scenarios, one of those might be easier to work with. [/QUOTE]
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Original definition of Riemann Integral and Darboux Sums
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