45. Let a> 0. Prove that f(X)= iS uniformly continuaus on on [a,00), So f(c1) = a. Similarly, there exists a c2 E (a, b] with f(c2) = B. %3D Definition. Let ECR and f a function with domain E. f is uniformly continuous on E if for all E> 0, there exists a 8(e) > 0 such that for all x, y E E with |x -y| < 8, |f(x) – f(y)| < e. Example. Let ECR be bounded. Then f(x) = x² is uniformly continuous on E.
45. Let a> 0. Prove that f(X)= iS uniformly continuaus on on [a,00), So f(c1) = a. Similarly, there exists a c2 E (a, b] with f(c2) = B. %3D Definition. Let ECR and f a function with domain E. f is uniformly continuous on E if for all E> 0, there exists a 8(e) > 0 such that for all x, y E E with |x -y| < 8, |f(x) – f(y)| < e. Example. Let ECR be bounded. Then f(x) = x² is uniformly continuous on E.
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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45
Transcribed Image Text:45. Let a> 0. Prove that f(X)= iS uniformly continuaus on
on [a,00),
![So f(c1) = a. Similarly, there exists a c2 E (a, b] with f(c2) = B.
%3D
Definition. Let ECR and f a function with domain E. f is uniformly continuous on E if for all
E> 0, there exists a 8(e) > 0 such that for all x, y E E with |x -y| < 8, |f(x) – f(y)| < e.
Example. Let ECR be bounded. Then f(x) = x² is uniformly continuous on E.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff82c0bab-f337-4781-985d-1029c419adb6%2Fac828b03-5a9d-4ccd-a622-50880b93838c%2Fh6co8os.jpeg&w=3840&q=75)
Transcribed Image Text:So f(c1) = a. Similarly, there exists a c2 E (a, b] with f(c2) = B.
%3D
Definition. Let ECR and f a function with domain E. f is uniformly continuous on E if for all
E> 0, there exists a 8(e) > 0 such that for all x, y E E with |x -y| < 8, |f(x) – f(y)| < e.
Example. Let ECR be bounded. Then f(x) = x² is uniformly continuous on E.
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