(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 12 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 12 2. Show that g is Riemann integrable on the entire interval [0, 1].
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
Related questions
Question
![(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 12 2. Show that g is Riemann integrable on the entire
interval [0, 1].
(Hint: Let € > 0 be given and let M > 0 be a constant such that g(z)| ≤ M for all z = [0, 1].
Choose n ≥ 2 so that < and note that
sup{9(2): x = [0,1/n]} - inf{9(x) : x = [0,1/n]} < ²5.
Now use that gis Riemann integrable on [1/m, 1] to find a suitable partition of [0, 1])
(c) Let / be a continuous function on [0, 1] and let g be the function given by
Jo,
when z = 0
[h(z), when 0<x< 1.
g(x) =
Show that g is Riemann integrable on [0, 1].
(Hint: Use the result from (b).)
JL](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9071221f-5482-42e2-a76d-06f20b84313c%2F421afc71-f0e9-4a25-abbe-792f4e3331f3%2Fny48p3m_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 12 2. Show that g is Riemann integrable on the entire
interval [0, 1].
(Hint: Let € > 0 be given and let M > 0 be a constant such that g(z)| ≤ M for all z = [0, 1].
Choose n ≥ 2 so that < and note that
sup{9(2): x = [0,1/n]} - inf{9(x) : x = [0,1/n]} < ²5.
Now use that gis Riemann integrable on [1/m, 1] to find a suitable partition of [0, 1])
(c) Let / be a continuous function on [0, 1] and let g be the function given by
Jo,
when z = 0
[h(z), when 0<x< 1.
g(x) =
Show that g is Riemann integrable on [0, 1].
(Hint: Use the result from (b).)
JL
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 23 images
Recommended textbooks for you
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
