- #1

Hall

- 351

- 81

$$

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?

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