(b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval [1/n, 1] is Riemann integrable for every n ≥2. Show that g is Riemann integrable on the entire interval [0, 1].
(b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval [1/n, 1] is Riemann integrable for every n ≥2. Show that g is Riemann integrable on the entire interval [0, 1].
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
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Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...
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![(b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval
[1/n, 1] is Riemann integrable for every n>2. Show that g is Riemann integrable on the entire
interval [0, 1].
(Hint: Let e > 0 be given and let M> 0 be a constant such that g(z)| ≤ M for all 2 = [0, 1].
Choose n ≥ 2 so that <and note that
2€
sup{g(z) : z = [0,1/n]} - inf{g(x) : z = [0,1/n]} < ²5
Now use that g is Riemann integrable on [1/n, 1] to find a suitable partition of [0, 1])](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8070592d-4a31-42d2-bef1-b46c168ad043%2F92d21fcc-7858-450c-80e5-9f49f24860d5%2Fuok9z4_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval
[1/n, 1] is Riemann integrable for every n>2. Show that g is Riemann integrable on the entire
interval [0, 1].
(Hint: Let e > 0 be given and let M> 0 be a constant such that g(z)| ≤ M for all 2 = [0, 1].
Choose n ≥ 2 so that <and note that
2€
sup{g(z) : z = [0,1/n]} - inf{g(x) : z = [0,1/n]} < ²5
Now use that g is Riemann integrable on [1/n, 1] to find a suitable partition of [0, 1])
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