Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a solution of the equation f(r) = x, r in R.
Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a solution of the equation f(r) = x, r in R.
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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![Suppose that f : R → R is continuous and that its image f(R) is bounded. Prove that there is
a solution of the equation
f (x) = x,
x in R.
(Hint: First assume |y| < M for any y E f(R). Then apply IVT for g(x) = f(x) – x on [-2M, 2M].)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F072e444d-1bde-4899-b3c3-9f07885f3d58%2Fa9781808-123b-4e88-9817-fa1923bf8a6d%2Fkr5oii9_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that f : R → R is continuous and that its image f(R) is bounded. Prove that there is
a solution of the equation
f (x) = x,
x in R.
(Hint: First assume |y| < M for any y E f(R). Then apply IVT for g(x) = f(x) – x on [-2M, 2M].)
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