44. Suppose that ƒ : [a, b] → [a, b] is continuous. Prove that there is at least one fixed point in [a, b]-that is, x such that f(x) = x. between f(u)
44. Suppose that ƒ : [a, b] → [a, b] is continuous. Prove that there is at least one fixed point in [a, b]-that is, x such that f(x) = x. between f(u)
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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Can you do 44 pl
![44. Suppose that ƒ : [a, b] → [a, b] is continuous. Prove that there is at least one fixed point in
[a, b]-that is, x such that f(x) = X.
45. If f: [a, b] →R is 1-1 and has the intermediate-value property-that is, if y is between f(u)
and f(u), there is x between u and v such that f(x) = y-show that f is continuous. (Hint:
First show that f is monotone.)
Prove
at there
16
tinuous function f RR such that, for each c ER, the equation
●](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F333b4161-3481-46d0-b492-9949c9218e0f%2F87jrmpu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:44. Suppose that ƒ : [a, b] → [a, b] is continuous. Prove that there is at least one fixed point in
[a, b]-that is, x such that f(x) = X.
45. If f: [a, b] →R is 1-1 and has the intermediate-value property-that is, if y is between f(u)
and f(u), there is x between u and v such that f(x) = y-show that f is continuous. (Hint:
First show that f is monotone.)
Prove
at there
16
tinuous function f RR such that, for each c ER, the equation
●
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