Let a, b € R. Let ƒ : [a, b] → R be an integrable function. Let g : [a,b] → R be a function which agrees with f at all points in [a, b] except for one, i.e. assume there exists a c € [a, b] such that g(x) = f(x) for all x € [a, b]\{c}. And g(c) < f(c). Prove that g is integrable on [a, b] and that fog(x)dx = So f(x)dx. Hint: Prove that g(x) − f(x) is integrable and the integral is 0.
Let a, b € R. Let ƒ : [a, b] → R be an integrable function. Let g : [a,b] → R be a function which agrees with f at all points in [a, b] except for one, i.e. assume there exists a c € [a, b] such that g(x) = f(x) for all x € [a, b]\{c}. And g(c) < f(c). Prove that g is integrable on [a, b] and that fog(x)dx = So f(x)dx. Hint: Prove that g(x) − f(x) is integrable and the integral is 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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![Let a, b € R. Let ƒ : [a, b] → R be an integrable function. Let g : [a,b] → R
be a function which agrees with f at all points in [a, b] except for one, i.e.
assume there exists a c € [a, b] such that g(x) = f(x) for all x € [a, b]\{c}.
And g(c) < f(c). Prove that g is integrable on [a, b] and that fog(x)dx =
So f(x)dx.
Hint: Prove that g(x) − f(x) is integrable and the integral is 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2F86470de4-e505-4a05-81d7-35aaeccbc5bb%2Fxvpwhk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a, b € R. Let ƒ : [a, b] → R be an integrable function. Let g : [a,b] → R
be a function which agrees with f at all points in [a, b] except for one, i.e.
assume there exists a c € [a, b] such that g(x) = f(x) for all x € [a, b]\{c}.
And g(c) < f(c). Prove that g is integrable on [a, b] and that fog(x)dx =
So f(x)dx.
Hint: Prove that g(x) − f(x) is integrable and the integral is 0.
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