Suppose f: [0, 1] → R and g: [0, 1] → Rare such that for all x E (0, 1), we have f(x) = g(x). Suppose f is Riemann integrable. f Prove that g is Riemann integrable and integral from 0 to 1 of f = integral from 0 to 1 g
Suppose f: [0, 1] → R and g: [0, 1] → Rare such that for all x E (0, 1), we have f(x) = g(x). Suppose f is Riemann integrable. f Prove that g is Riemann integrable and integral from 0 to 1 of f = integral from 0 to 1 g
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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Can someone please rewrite this proof? The question is below:
Suppose f: [0, 1] → R and g: [0, 1] → Rare such that for all x E (0, 1), we have f(x) = g(x). Suppose f is Riemann
![det
sum,
m
n mbintenab
M-1
nd, 1} and consider the right Riemann
ni
n
R₁ = = f(h)
1=1
== O
Now,
2
{ 0, h = 1,...,
n/
partition [0,1] inta
Hence,
Lince,
f, no,
and,
-
since, him. Premann integrable
1
Show da = lim Rn = 0
b
ma
= 0
h in Riemann integrable
f-h is also Riemann integrable
Sit-us
=
on [0, 1],
St
on
S's - S'u
f
h
0
in Riemann integrable on
S'o = 1/s
0
[0,1] and no is
on [0,1]
[0, 1] and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff12a5893-6ffe-4c09-825d-e62643fd26aa%2Fc86baf94-65fb-4cf2-a723-4d28f178562d%2Fqystdtv_processed.png&w=3840&q=75)
Transcribed Image Text:det
sum,
m
n mbintenab
M-1
nd, 1} and consider the right Riemann
ni
n
R₁ = = f(h)
1=1
== O
Now,
2
{ 0, h = 1,...,
n/
partition [0,1] inta
Hence,
Lince,
f, no,
and,
-
since, him. Premann integrable
1
Show da = lim Rn = 0
b
ma
= 0
h in Riemann integrable
f-h is also Riemann integrable
Sit-us
=
on [0, 1],
St
on
S's - S'u
f
h
0
in Riemann integrable on
S'o = 1/s
0
[0,1] and no is
on [0,1]
[0, 1] and
![Let, hi [0, 1] → R defined by h(x) = fexs-g(x) *x.
Then, hex) = { 0, KE (0,1]
fco)-g(0), x = 0
Hence or in discontinuous on
fraint in [0,1].
Hence, using Lebesgue's Integrability Criterion hi
Riemann integrable on [0, 1].
atmast
one](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff12a5893-6ffe-4c09-825d-e62643fd26aa%2Fc86baf94-65fb-4cf2-a723-4d28f178562d%2Frjeh32b_processed.png&w=3840&q=75)
Transcribed Image Text:Let, hi [0, 1] → R defined by h(x) = fexs-g(x) *x.
Then, hex) = { 0, KE (0,1]
fco)-g(0), x = 0
Hence or in discontinuous on
fraint in [0,1].
Hence, using Lebesgue's Integrability Criterion hi
Riemann integrable on [0, 1].
atmast
one
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