Let f: [0, 1] → [0, 1] be a continuous strictly increasing function.(a) If f is absolutely continuous on [0, 1], prove that for any open set UC [0, 1] we have\ƒ(U)| = √√ ƒ'(x) dx .SoU(b) Give an example that (1) is not true without the assumption of absolute continuity.

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Answered: Let f: [0, 1] → [0, 1] be a continuous…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

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Let f: [0, 1] → [0, 1] be a continuous strictly increasing function. (a) If ƒ is absolutely continuous on [0, 1], prove that for any open set UC [0, 1] we have \ƒ(U)] = √√ f'(x) dx . (b) Give an example that (1) is not true without the assumption of absolute continuity.