2.50 Prove: A function f is bounded if and only if || f |lsup < o. (Hint: See Definition 2.4.1.) 2.4 EXTREME VALUE THEOREM AND ITS CONSEQUENCES Definition 2.4.1 We say a function f is bounded on a domain D, provided that there exist m and M in R such that m < f(x) < M for all x E D. Observe that f(x) = x is continuous on R but f is not bounded on R. And g(x) = ! is continuous on (0,1) but is not bounded, since it lacks an upper bound, though there is a lower bound, 0. We have, however, the following important theorem pertaining to continuous functions on closed, finite intervals.
2.50 Prove: A function f is bounded if and only if || f |lsup < o. (Hint: See Definition 2.4.1.) 2.4 EXTREME VALUE THEOREM AND ITS CONSEQUENCES Definition 2.4.1 We say a function f is bounded on a domain D, provided that there exist m and M in R such that m < f(x) < M for all x E D. Observe that f(x) = x is continuous on R but f is not bounded on R. And g(x) = ! is continuous on (0,1) but is not bounded, since it lacks an upper bound, though there is a lower bound, 0. We have, however, the following important theorem pertaining to continuous functions on closed, finite intervals.
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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![2.50 Prove: A function f is bounded if and only if ||filsup < ∞. (Hint: See
Definition 2.4.1.)
2.4 EXTREME VALUE THEOREM AND ITS CONSEQUENCES
Definition 2.4.1 We say a function f is bounded on a domain D, provided that
there exist m and M in R such that
m< f(x) < M
for all x E D.
Observe that f (x) = x is continuous on R but f is not bounded on R. And g(x) - !
is continuous on (0,1) but is not bounded, since it lacks an upper bound, though there
is a lower bound, 0. We have, however, the following important theorem pertaining
to continuous functions on closed, finite intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe87a569b-7721-4104-9b29-fedcf0a2817b%2Fc61792b9-2152-4fc1-918c-cc7dab156d4b%2F9f2ta1t_processed.png&w=3840&q=75)
Transcribed Image Text:2.50 Prove: A function f is bounded if and only if ||filsup < ∞. (Hint: See
Definition 2.4.1.)
2.4 EXTREME VALUE THEOREM AND ITS CONSEQUENCES
Definition 2.4.1 We say a function f is bounded on a domain D, provided that
there exist m and M in R such that
m< f(x) < M
for all x E D.
Observe that f (x) = x is continuous on R but f is not bounded on R. And g(x) - !
is continuous on (0,1) but is not bounded, since it lacks an upper bound, though there
is a lower bound, 0. We have, however, the following important theorem pertaining
to continuous functions on closed, finite intervals.
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