3. Let A be the set of all functions ƒ : 0, 1] → R that are continuous in the interval [0, 1]. Recall that I : A → R defined as I(f) = | f(x)dx is a function. (a) Prove that I is surjective. (b) Show that I is not injective.
3. Let A be the set of all functions ƒ : 0, 1] → R that are continuous in the interval [0, 1]. Recall that I : A → R defined as I(f) = | f(x)dx is a function. (a) Prove that I is surjective. (b) Show that I is not injective.
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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![3. Let A be the set of all functions f: [0, 1] →R that are continuous in the interval [0, 1].
Recall that I : A → R defined as
I(f) = | f(x)dx
is a function.
(a) Prove that I is surjective.
(b) Show that I is not injective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36c20647-fb28-4322-9604-6482a64aaa32%2F551481ee-edb4-4380-ba3c-b5ecfce189de%2Fthjvg9m_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let A be the set of all functions f: [0, 1] →R that are continuous in the interval [0, 1].
Recall that I : A → R defined as
I(f) = | f(x)dx
is a function.
(a) Prove that I is surjective.
(b) Show that I is not injective.
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