Suppose f : [a, b] → R is bounded and for each ε > 0 there is a partition P such that forany refinements Q1 and Q2 of P , regardless of how marked, |S(Q1, f ) − S(Q2, f )| < ε.Prove that f is integrable on [a, b].

Suppose f : [a, b] → R is bounded and for each ε > 0 there is a partition P such that forany refinements Q1 and Q2 of P , regardless of how marked, |S(Q1, f ) − S(Q2, f )| < ε.Prove that f is integrable on [a, b].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 16E: Prove that if a subring R of an integral domain D contains the unity element of D, then R is an...

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Question

Suppose f : [a, b] → R is bounded and for each ε > 0 there is a partition P such that for
any refinements Q1 and Q2 of P , regardless of how marked, |S(Q1, f ) − S(Q2, f )| < ε.
Prove that f is integrable on [a, b].

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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