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" Riemann integrable over [a, b] if and only if for every e > 0, there exists... "
Real Analysis - Seite 220
von Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson - 1997 - 713 Seiten
Vollansicht - Über dieses Buch

Integral: An Easy Approach After Kurzweil and Henstock

Lee Peng Yee, Rudolf Vyborny - 2000 - 328 Seiten
...not AC*(X) where X is a closed subset of [a, 6]. EXERCISE 5.2 Show that F is absolutely continuous on [a, b] if and only if for every e > 0 there exists 77 > 0 such that for every sequence {[a,, bi]} of non-overlapping intervals with Y^Li l6i ~ °t| <...
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Understanding Analysis

Stephen Abbott - 2002 - 284 Seiten
...upper and lower sums are arbitrarily close together. Theorem 7.2.8. A bounded function f is integrable on [a, b] if and only if, for every e > 0, there exists a partition Pf of [a, b] such that Proof. Let e > 0. If such a partition Pf exists, then U(f)-L(f)<U(f,P.)-L(f,Pt)<f....
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Computer-Supported Calculus

Adi Ben-Israel, Robert P. Gilbert - 2002 - 582 Seiten
...f(x)dx. Remark 8.18. The above results can be stated formally as follows: The function / is integrable on [a, b] if, and only if, for every e > 0 there is a partition 3* of [a, b] such that Su(/, 3*) - SL(/, P) < e . We define the approximation error...
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Finite-Dimensional Variational Inequalities and Complementarity Problems

Francisco Facchinei, Jong-Shi Pang - 2007 - 693 Seiten
...K is compact. (c) Suppose that SOL(-ft', F) is a finite set. Show that the VI (K, F) is semistable if and only if for every e > 0, there exists 5 > 0, such that || 9 || < S => SOL(K,q+F) C SOL(K, F) + B(0, e). 5.6.19 Let C CM™ 1 be a pointed, solid,...
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A Concise Approach to Mathematical Analysis

Mangatiana A. Robdera - 2003 - 390 Seiten
...such that it is the limit of some sequence in A, and let I be a real number. Then limx^ax / (x) = I if and only if for every e > 0, there exists <5 > 0 such that , , , , (41) x € A and \x - a\ < 6 implies |/ (x) - l\ < e. Proof Suppose that (4.1) holds...
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