(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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
c)
![(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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f3f25b-2eb2-4186-b342-f9f487e8326d%2Ffad42919-3cee-4bd1-9ebb-cc594d708bdd%2F7dtfhv_processed.png&w=3840&q=75)
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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