Let a < b and D : [a, b] → R, be defined by 0 if D(x) = { 1 if xe (R\Q) x € Q Show that D is not Riemann integrable. = Hint: Show that for any partition P of [a, b], U (f, P) = ba and L(f, P) = choose € = ba > 0, then for all partition P of [a, b], U(ƒ, P) – L(ƒ, P) = b − a > €. Therefore, if we
Let a < b and D : [a, b] → R, be defined by 0 if D(x) = { 1 if xe (R\Q) x € Q Show that D is not Riemann integrable. = Hint: Show that for any partition P of [a, b], U (f, P) = ba and L(f, P) = choose € = ba > 0, then for all partition P of [a, b], U(ƒ, P) – L(ƒ, P) = b − a > €. Therefore, if we
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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Please include a picture of your solution and answer all parts clearly. Thanks
![Let a < b and D : [a, b] → R, be defined by
0
if
{
if x EQ
D (x)
=
1
x = (R\Q)
Show that D is not Riemann integrable.
Hint: Show that for any partition P of [a, b], U(f, P) = b − a and L(f, P) = 0. Therefore, if we
choose € = b-a > 0, then for all partition P of [a, b],
2
U (ƒ, P) – L(ƒ, P) = b − a > €.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F2b3af11d-2190-480e-a4ed-1a4067b42299%2Fdhe7mea_processed.png&w=3840&q=75)
Transcribed Image Text:Let a < b and D : [a, b] → R, be defined by
0
if
{
if x EQ
D (x)
=
1
x = (R\Q)
Show that D is not Riemann integrable.
Hint: Show that for any partition P of [a, b], U(f, P) = b − a and L(f, P) = 0. Therefore, if we
choose € = b-a > 0, then for all partition P of [a, b],
2
U (ƒ, P) – L(ƒ, P) = b − a > €.
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