Mathquery

Darboux's Theorem For Integrability :           If  f is a bounded function on [a,b] , then to every ε > 0 , there corresponds         δ > 0, such that                         -b  (A)   U(P,f) < ∫ f dx + ε                           a                                                          b   (B)    L(P,f) > ∫ f dx - ε                        - a for every partition P of [ a,b ] with norm μ(P) < δ . Proof : (A)        Given f is bounded function of [a,b] , so there exists a positive number k          such that          |f(x) | ≤ k   ∀ x ∈ [ a,b]    We know the upper integral is the infimum of the set of upper sums i.e to every ε > 0 , ∃ a partition P₁ = { x₀,x₁,x₂,.....xₙ}  of [ a,b ] such that                   - b     U(P,f) < ∫ f dx + ε / 2 ..........(1)                   a Here the partition P has p-1 points besides x₀ and xₚ . Let δ be a positive number     such that          2 k (p-1) δ = ε

Conditions Of Integrability :    ( A ) First Form Of Integrability :                   A necessary and sufficient condition for the integrability of a bounded function f is that to every ε > 0 , there corresponds δ > 0 such that for every partition P of [a,b] with norms μ(P) < δ ,   U(P,f) - L(P,f) < ε . Proof Of Necessary Part :               Let's suppose that the bounded function f be integrable i.e          b          - b            b        ∫ f dx = ∫ f dx = ∫ f dx        -a            a             a      Let ε > 0 be a positive small number . By Darboux's Theorem ∃ a δ > 0 such that for every partition P with norm μ(P) < δ we have                             - b                         b              U(P,f) < ∫  f dx + ε/2 = ∫ f dx + ε /2 ...(1)                             a                         a            and                              b                     b              L(P,f) > ∫