11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on (a, b]. 'x→ 12. The converse to Problem 11 is also true. If f is uniformly continuous on (a,b], then lim,at f(x) exists. We outline a proof. (a) Suppose f is uniformly continuous on (a,b], and let {x,} be any fixed sequence in (a, b] converging to a. Show that the sequence {f(xn)} has a convergent subsequence. (Hint: Use the Bolzano-Weierstrass Theorem.) (b) Let L be the limit of the convergent subsequence of part (a). Prove, using the uniform continuity of f on (a, b], that lim,x¬a* (x) = L. ||
11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on (a, b]. 'x→ 12. The converse to Problem 11 is also true. If f is uniformly continuous on (a,b], then lim,at f(x) exists. We outline a proof. (a) Suppose f is uniformly continuous on (a,b], and let {x,} be any fixed sequence in (a, b] converging to a. Show that the sequence {f(xn)} has a convergent subsequence. (Hint: Use the Bolzano-Weierstrass Theorem.) (b) Let L be the limit of the convergent subsequence of part (a). Prove, using the uniform continuity of f on (a, b], that lim,x¬a* (x) = L. ||
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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12 (b)
![11. Prove that if f is continuous on (a, b] and lim, a+ f(x) exists,
then fis uniformly continuous on (a, b].
X→a
12. The converse to Problem 11 is also true. If f is uniformly
continuous on (a,b], then limy»a+ f(x) exists. We outline a proof.
(a) Suppose f is uniformly continuous on (a,b], and let {x,} be
any fixed sequence in (a, b] converging to a. Show that the
sequence {f(xn)} has a convergent subsequence. (Hint: Use
the Bolzano-Weierstrass Theorem.)
(b) Let L be the limit of the convergent subsequence of part (a).
Prove, using the uniform continuity of f on (a, b], that lim,a*
f(x) = L.
X→a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ab050a-d13e-4a73-9fee-dc44d5d071e8%2F9072a861-63c8-49ad-87cf-a27bee599e2e%2F5y6b2kk_processed.png&w=3840&q=75)
Transcribed Image Text:11. Prove that if f is continuous on (a, b] and lim, a+ f(x) exists,
then fis uniformly continuous on (a, b].
X→a
12. The converse to Problem 11 is also true. If f is uniformly
continuous on (a,b], then limy»a+ f(x) exists. We outline a proof.
(a) Suppose f is uniformly continuous on (a,b], and let {x,} be
any fixed sequence in (a, b] converging to a. Show that the
sequence {f(xn)} has a convergent subsequence. (Hint: Use
the Bolzano-Weierstrass Theorem.)
(b) Let L be the limit of the convergent subsequence of part (a).
Prove, using the uniform continuity of f on (a, b], that lim,a*
f(x) = L.
X→a
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