A fundamental fact is that if we are working on a closed, bounded inter [a, b], then any continuous function f : [a, b] → R is automatically uniform continuous. Theorem. Suppose f [a, b] → R is continuous. Then it is uniform continuous. 9. Find an example of an ƒ : (0, 1) → R which is continous but not uniform continuous. Where exactly did we use the fact that [a, b] was a closed a bounded interval in the proof of the theorem?
A fundamental fact is that if we are working on a closed, bounded inter [a, b], then any continuous function f : [a, b] → R is automatically uniform continuous. Theorem. Suppose f [a, b] → R is continuous. Then it is uniform continuous. 9. Find an example of an ƒ : (0, 1) → R which is continous but not uniform continuous. Where exactly did we use the fact that [a, b] was a closed a bounded interval in the proof of the theorem?
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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![A fundamental fact is that if we are working on a closed, bounded interval
[a, b], then any continuous function f [a, b] → R is automatically uniformly
continuous.
Theorem. Suppose f [a, b] → R is continuous. Then it is uniformly
continuous.
9. Find an example of an ƒ : (0, 1) → R which is continous but not uniformly
continuous. Where exactly did we use the fact that [a, b] was a closed and
bounded interval in the proof of the theorem?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2F999c8b73-d0e3-4519-bad9-9de3f94b968a%2F1sxeo4m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A fundamental fact is that if we are working on a closed, bounded interval
[a, b], then any continuous function f [a, b] → R is automatically uniformly
continuous.
Theorem. Suppose f [a, b] → R is continuous. Then it is uniformly
continuous.
9. Find an example of an ƒ : (0, 1) → R which is continous but not uniformly
continuous. Where exactly did we use the fact that [a, b] was a closed and
bounded interval in the proof of the theorem?
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