A function f is of bounded variation on the closed interval [a, b] if there is a number K such that m Eis(=;) – f(xj-1)| < K j=1 for any partition P = {a = xo < x1 < • ·· < xm = b}. Assume f is of bounded variation on [a, b]. Prove the following statements: (i) The function f is bounded on [a, b]. (ii) The function f is integrable on [a,b].
A function f is of bounded variation on the closed interval [a, b] if there is a number K such that m Eis(=;) – f(xj-1)| < K j=1 for any partition P = {a = xo < x1 < • ·· < xm = b}. Assume f is of bounded variation on [a, b]. Prove the following statements: (i) The function f is bounded on [a, b]. (ii) The function f is integrable on [a,b].
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 function f is of bounded variation on the closed interval [a, b] if there is a number K
such that
Els(2;) – f(xj–1)| < K
j=1
for any partition P = {a = xo < x1 < • · · < xm = b}. Assume f is of bounded variation
on [a, b]. Prove the following statements:
%3D
(i) The function ƒ is bounded on [a, b].
(ii) The function f is integrable on [a,b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d962b91-aae5-4848-8637-6da496958574%2F5e22cadc-d0b1-40ad-bd0a-8ec8b02f8e9e%2F9jbt78p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A function f is of bounded variation on the closed interval [a, b] if there is a number K
such that
Els(2;) – f(xj–1)| < K
j=1
for any partition P = {a = xo < x1 < • · · < xm = b}. Assume f is of bounded variation
on [a, b]. Prove the following statements:
%3D
(i) The function ƒ is bounded on [a, b].
(ii) The function f is integrable on [a,b].
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