(a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0
(a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0
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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![(Rudin, Ch. 6, Exercise 7) Suppose f : (0, 1]→ R is Riemann integrable over [c, 1] for every c> 0.
(a) Show that if f is Riemann integrable over [0, 1], then
| S(x) dr = lim
c→0
(b) Construct a function f satisfying the hypotheses such that
lim
c→0
f(x) dx
exists
but
lim
c→0
/ If(x)| dx
does not exist.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a8e1c4f-5637-49d4-a073-063bf535e94b%2F515c752e-8c0e-4bfd-83a3-57178f2809d1%2Fka1d4uq_processed.png&w=3840&q=75)
Transcribed Image Text:(Rudin, Ch. 6, Exercise 7) Suppose f : (0, 1]→ R is Riemann integrable over [c, 1] for every c> 0.
(a) Show that if f is Riemann integrable over [0, 1], then
| S(x) dr = lim
c→0
(b) Construct a function f satisfying the hypotheses such that
lim
c→0
f(x) dx
exists
but
lim
c→0
/ If(x)| dx
does not exist.
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