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Original definition of Riemann Integral and Darboux Sums
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[QUOTE="Hall, post: 6839587, member: 696593"] Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a ##\delta_{\epsilon} \gt 0 ## such that, $$ || \dot P|| \lt \delta_{\epsilon} \implies | S(f, \dot P) - L| \lt \varepsilon$$. And we write, ##\int_{a}^{b} f = L## The problem is we don't know ##L## before-hand, so how are we suppose to carry out the epsilon-delta process? The Darboux Sums allow us to define Upper Integral and Lower Integral as follows: $$ U(f) = \inf \{ U(f,P) : P \in \mathcal{P} \}$$ $$ L(f) = \sup \{ L(f,P) : P \in \mathcal{P} \}$$ And a function is integrable if ##U(f) = L (f)##. This Darboux Sum and Integral have two major advantages: the existence of infimum and supremum is guaranteed by the Completeness Axiom, and Upper and Lower sums can easily be calculated. We define the integral ##\int_{a}^{b} f = U(f) = L (f)##. Though, we can establish the equivalence relation between Riemann's original definition and Darboux's definition, but how do we, in practice, use Riemann's definition for proving the integrability of a function? And if Darboux's definition was more practical then why was there a need for Riemann's definition? What did Darboux's formulation lack? ˙ [/QUOTE]
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Original definition of Riemann Integral and Darboux Sums
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