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(b) Restricted Function: Let g be a bounded function on [0, 1] and assume that its restriction to the interval [1/n, 1] is Riemann integrable for every n > 2. You are to show that g is Riemann integrable on the entire interval [0, 1]. Hint: You're given to let € > 0 and M > 0 be constants such that g(x)| ≤M for all a in [0, 1]. Choose n > 2 such that <3 and note that the supremum and infimum of g on [0, 1/n] differ by less than 2. You are to show that g is Riemann integrable on [1/n, 1] and find an appropriate partition of [0, 1]. (c) Continuous Function: Leth be a continuous function on [0, 1] and let g be the function defined by: g(x) = Hint: Use the result from (b). for x = 0 h(x) for 0

(b) Restricted Function: Let g be a bounded function on [0, 1] and assume that its restriction to the interval [1/n, 1] is Riemann integrable for every n > 2. You are to show that g is Riemann integrable on the entire interval [0, 1]. Hint: You're given to let € > 0 and M > 0 be constants such that g(x)| ≤M for all a in [0, 1]. Choose n > 2 such that <3 and note that the supremum and infimum of g on [0, 1/n] differ by less than 2. You are to show that g is Riemann integrable on [1/n, 1] and find an appropriate partition of [0, 1]. (c) Continuous Function: Leth be a continuous function on [0, 1] and let g be the function defined by: g(x) = Hint: Use the result from (b). for x = 0 h(x) for 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(b) Restricted Function: Let g be a bounded function on [0, 1] and assume that its restriction to the interval [1/n, 1] is Riemann integrable for every n > 2. You are to show that g is Riemann integrable on the entire interval [0, 1]. Hint: You're given to let € > 0 and M > 0 be constants such that g(x)| ≤M for all a in [0, 1]. Choose n > 2 such that <3 and note that the supremum and infimum of g on [0, 1/n] differ by less than 2. You are to show that g is Riemann integrable on [1/n, 1] and find an appropriate partition of [0, 1]. (c) Continuous Function: Leth be a continuous function on [0, 1] and let g be the function defined by: g(x) = Hint: Use the result from (b). for x = 0 h(x) for 0<x<1 You are asked to show that g is Riemann integrable on [0, 1].
Transcribed Image Text:(b) Restricted Function: Let g be a bounded function on [0, 1] and assume that its restriction to the interval [1/n, 1] is Riemann integrable for every n > 2. You are to show that g is Riemann integrable on the entire interval [0, 1]. Hint: You're given to let € > 0 and M > 0 be constants such that g(x)| ≤M for all a in [0, 1]. Choose n > 2 such that <3 and note that the supremum and infimum of g on [0, 1/n] differ by less than 2. You are to show that g is Riemann integrable on [1/n, 1] and find an appropriate partition of [0, 1]. (c) Continuous Function: Leth be a continuous function on [0, 1] and let g be the function defined by: g(x) = Hint: Use the result from (b). for x = 0 h(x) for 0<x<1 You are asked to show that g is Riemann integrable on [0, 1].
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