Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does the answer change if we assume that f ER?
Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does the answer change if we assume that f ER?
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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Q5 Real analysis by Walton rudin "Riemann stieltjes integral "
![|Tr()| dt < A(P, y) + 2e(b – a)
< A(y) + 28(b – a).
Since e was arbitrary,
dt < A(y).
This completes the proof.
138 PRINCIPLES OF MATHEMATICAL ANALYSIS
EXERCISES
1. Suppose a increases on [a, b), asxo sb, a is continuous at xo, f(xo)=1, and
f(x) = 0 if x xo. Prove that fe R(x) and that ff da = 0.
2. Suppose f20, ƒ is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0
for all x e [a, b). (Compare this with Exercise 1.)
3. Define three functions B1, B2, B3 as follows: B,(x) = 0 if x<0, B,(x) =1 if x>0
for j= 1, 2, 3; and B,(0) = 0, B2(0) =1, B3(0) = 1. Let ƒ be a bounded function on
(-1, 1).
(a) Prove that fe R(B,) if and only if f(0+) = f(0) and that then
=f(0).
(b) State and prove a similar result for B2.
(c) Prove that fe R(B,) if and only if f is continuous at 0.
(d) If f is continuous at 0 prove that
4. If f(x) = 0 for all irrational x, f(x) = 1 for all rational x, prove that f¢ R on[a, b}
for any a <b.
5. Suppose f is a bounded real function on [a, b], and f' eR on [a, b]. Does it
follow that fe R? Does the answer change if we assume that f'e R?
6. Let P be the Cantor set constructed in Sec. 2.44. Let f be a bounded real function
on (0, 1] which is continuous at every point outside P. Prove that fe R on [0, 1].
Hint: P can be covered by finitely many segments whose total length can be made
as small as desired. Proceed as in Theorem 6.10.
7. Suppose f is a real function on (0, 1] and fe R on [c, 1] for every c>0. Define
( (x) dx= lim
if this limit exists (and is finite).
(a) If fe R on [0, 1), show that this definition of the integral agrees with the old
one.
(b) Construct a function f such that the above limit exists, although it fails to exist
with | f| in place of f.
8. Suppose fe R on [a, b] for every b>a where a is fixed. Define
( f(x) dx= lim
if this limit exists (and is finite). In that case, we say that the integral on the left
converges. If it also converges after f has been replaced by f, it is said to con-
verge absolutely.
THE RIEMANN-STIELTJES INTEGRAL
139
Assume that f(x) 20 and that f decreases monotonically on [1, 0). Prove
that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa29dd1fd-ff33-4092-aedd-03e5d354d4ee%2Fcd7adeea-521a-4b2c-a3e9-10ff6f8012d2%2Fij5sqql_processed.jpeg&w=3840&q=75)
Transcribed Image Text:|Tr()| dt < A(P, y) + 2e(b – a)
< A(y) + 28(b – a).
Since e was arbitrary,
dt < A(y).
This completes the proof.
138 PRINCIPLES OF MATHEMATICAL ANALYSIS
EXERCISES
1. Suppose a increases on [a, b), asxo sb, a is continuous at xo, f(xo)=1, and
f(x) = 0 if x xo. Prove that fe R(x) and that ff da = 0.
2. Suppose f20, ƒ is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0
for all x e [a, b). (Compare this with Exercise 1.)
3. Define three functions B1, B2, B3 as follows: B,(x) = 0 if x<0, B,(x) =1 if x>0
for j= 1, 2, 3; and B,(0) = 0, B2(0) =1, B3(0) = 1. Let ƒ be a bounded function on
(-1, 1).
(a) Prove that fe R(B,) if and only if f(0+) = f(0) and that then
=f(0).
(b) State and prove a similar result for B2.
(c) Prove that fe R(B,) if and only if f is continuous at 0.
(d) If f is continuous at 0 prove that
4. If f(x) = 0 for all irrational x, f(x) = 1 for all rational x, prove that f¢ R on[a, b}
for any a <b.
5. Suppose f is a bounded real function on [a, b], and f' eR on [a, b]. Does it
follow that fe R? Does the answer change if we assume that f'e R?
6. Let P be the Cantor set constructed in Sec. 2.44. Let f be a bounded real function
on (0, 1] which is continuous at every point outside P. Prove that fe R on [0, 1].
Hint: P can be covered by finitely many segments whose total length can be made
as small as desired. Proceed as in Theorem 6.10.
7. Suppose f is a real function on (0, 1] and fe R on [c, 1] for every c>0. Define
( (x) dx= lim
if this limit exists (and is finite).
(a) If fe R on [0, 1), show that this definition of the integral agrees with the old
one.
(b) Construct a function f such that the above limit exists, although it fails to exist
with | f| in place of f.
8. Suppose fe R on [a, b] for every b>a where a is fixed. Define
( f(x) dx= lim
if this limit exists (and is finite). In that case, we say that the integral on the left
converges. If it also converges after f has been replaced by f, it is said to con-
verge absolutely.
THE RIEMANN-STIELTJES INTEGRAL
139
Assume that f(x) 20 and that f decreases monotonically on [1, 0). Prove
that
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