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7. Let f: IR be a uniformly continuous map on a bounded set I. Show that f is bounded on I. u need not

7. Let f: IR be a uniformly continuous map on a bounded set I. Show that f is bounded on I. u need not

Advanced Engineering Mathematics
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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7. Let f: IR be a uniformly continuous map on a bounded set I. Show that f is bounded on I. 8. Give an example to show that a bounded continuous function on a bounded interval need not be uniformly continuous. 9. Let I be a an interval and f: IR be uniformly continuous. Let {n} be a sequence in I. Show that if {f(n)} is a Cauchy sequence, then {r) is a Cauchy sequence.
Transcribed Image Text:7. Let f: IR be a uniformly continuous map on a bounded set I. Show that f is bounded on I. 8. Give an example to show that a bounded continuous function on a bounded interval need not be uniformly continuous. 9. Let I be a an interval and f: IR be uniformly continuous. Let {n} be a sequence in I. Show that if {f(n)} is a Cauchy sequence, then {r) is a Cauchy sequence.
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