Let f [a, b] R and ro € (a, b). Assume that f is continuous on [a, b]\ {xo} and limx→zo f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into two parts. • First we show that f is bounded. - Since limx→ro f (x) = L, there is a d with 0 < d < min {xo-a, b-ro} such that f (x) - L|| < 1 whenever 0 < x-xo|< 8. - Show that f(x)| ≤ 1+ |L| whenever 0 < x-xo| ≤ Explain why and exist. Show that x sup {\ƒ (2)| : 2 € [a, xo - 2]} M₂ = sup {|ƒ (a) | : x € [xo + 2, b]} M₁ = sup{|ƒ |f (x)| ≤ 1+ |L| + |ƒ (xo)| + M₁ + M₂ for all x € [a, b]. • Now we show that given € > 0, there is a partition P of [a, b] such that U (f, P) – L(f, P) < €. - Since lim xo f (x) = L, there is a d with: Show that 0 < 6 < minxo-a, b-xo, ¹ {20 € 3 max{1+ f(xo) - L, 2} such that f (x) - L| < 1 whenever 0 < x-xo|< 6. Show that 8 sup {1 (2): 2 € [20-2,20 + ]}-inf{1(2): -inf f (x): x € To € [20-₁²0+]} ≤ ≤ max {1+ f(xo) - L, 2} - Explain why there is a partition P₁ of [a, xo] such that U (ƒ, P₁) — L (ƒ, P₁) < §. - Explain why there is a partition P2 of [xo + 2,6] such that U (f, P₂) - L (f, P₂) <. Consider the partition P = P₁UP₂ of [a, b], show that U (f, P) – L (f, P) < €.
Let f [a, b] R and ro € (a, b). Assume that f is continuous on [a, b]\ {xo} and limx→zo f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into two parts. • First we show that f is bounded. - Since limx→ro f (x) = L, there is a d with 0 < d < min {xo-a, b-ro} such that f (x) - L|| < 1 whenever 0 < x-xo|< 8. - Show that f(x)| ≤ 1+ |L| whenever 0 < x-xo| ≤ Explain why and exist. Show that x sup {\ƒ (2)| : 2 € [a, xo - 2]} M₂ = sup {|ƒ (a) | : x € [xo + 2, b]} M₁ = sup{|ƒ |f (x)| ≤ 1+ |L| + |ƒ (xo)| + M₁ + M₂ for all x € [a, b]. • Now we show that given € > 0, there is a partition P of [a, b] such that U (f, P) – L(f, P) < €. - Since lim xo f (x) = L, there is a d with: Show that 0 < 6 < minxo-a, b-xo, ¹ {20 € 3 max{1+ f(xo) - L, 2} such that f (x) - L| < 1 whenever 0 < x-xo|< 6. Show that 8 sup {1 (2): 2 € [20-2,20 + ]}-inf{1(2): -inf f (x): x € To € [20-₁²0+]} ≤ ≤ max {1+ f(xo) - L, 2} - Explain why there is a partition P₁ of [a, xo] such that U (ƒ, P₁) — L (ƒ, P₁) < §. - Explain why there is a partition P2 of [xo + 2,6] such that U (f, P₂) - L (f, P₂) <. Consider the partition P = P₁UP₂ of [a, b], show that U (f, P) – L (f, P) < €.
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 answer all parts of the question and show the proofs clearly. Thanks
![Let f [a, b] R and ro € (a, b). Assume that f is continuous on [a, b]\{o} and
limx→xo f (x) = L (L is finite) exists. Show that f is Riemann integrable.
Hint: We split it into two parts.
• First we show that f is bounded.
Since limx→ro f (x) = L, there is a d with
0 < 8 < min {xo-a, b-ro}
such that f (x) - L|| < 1 whenever 0 < x-xo| <d.
Show that f(x)| ≤ 1+|L| whenever 0 < x − xo| ≤ 1/2.
- Explain why
and
exist.
Show that
sup
{ƒ(x) = x €
M₁ = sup
p{\ƒ (x) \: x € [ª,‚ x₁ − ² ] }
xo-
for all x € [a, b].
• Now we show that given € > 0, there is a partition P of [a, b] such that U (f, P) – L (ƒ, P) < €.
- Since limx→xo f (x) = L, there is a d with: Show that
M₂ = sup
•{\ƒ (2)| : 2 € [x₁ + 2/₂ b]}
f
xo
: xx0
|ƒ (x)| ≤ 1 + |L| + |ƒ (xo)| + M₁ + M₂
0 < 6 < mino-
¹ {²₁ - α
8
10-₁10 +
such that f (x) - L| < 1 whenever 0 < x-xo|< 8.
Show that
a, b.
- x0,
-]}-inf {1 (2)
-inf f (x): x xo
1 € TO
€
3 max{1+ f(xo).
2) — 4₁, 2}}
L,
8
xo +
1}
<max {1+ f(xo) — L\,2}.
Explain why there is a partition P₁ of [a, xo] such that U (f, P₁) — L (ƒ, P₁) < §.
- Explain why there is a partition P₂ of [xo+ ,b] such that U (ƒ, P₂) — L (ƒ, P₂) < §.
Consider the partition P = P₁UP₂ of [a, b], show that U (f, P) – L (ƒ, P) < €.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F6902ed25-d83f-4996-b063-9dc89ed7cbbe%2Fg9rzmit_processed.png&w=3840&q=75)
Transcribed Image Text:Let f [a, b] R and ro € (a, b). Assume that f is continuous on [a, b]\{o} and
limx→xo f (x) = L (L is finite) exists. Show that f is Riemann integrable.
Hint: We split it into two parts.
• First we show that f is bounded.
Since limx→ro f (x) = L, there is a d with
0 < 8 < min {xo-a, b-ro}
such that f (x) - L|| < 1 whenever 0 < x-xo| <d.
Show that f(x)| ≤ 1+|L| whenever 0 < x − xo| ≤ 1/2.
- Explain why
and
exist.
Show that
sup
{ƒ(x) = x €
M₁ = sup
p{\ƒ (x) \: x € [ª,‚ x₁ − ² ] }
xo-
for all x € [a, b].
• Now we show that given € > 0, there is a partition P of [a, b] such that U (f, P) – L (ƒ, P) < €.
- Since limx→xo f (x) = L, there is a d with: Show that
M₂ = sup
•{\ƒ (2)| : 2 € [x₁ + 2/₂ b]}
f
xo
: xx0
|ƒ (x)| ≤ 1 + |L| + |ƒ (xo)| + M₁ + M₂
0 < 6 < mino-
¹ {²₁ - α
8
10-₁10 +
such that f (x) - L| < 1 whenever 0 < x-xo|< 8.
Show that
a, b.
- x0,
-]}-inf {1 (2)
-inf f (x): x xo
1 € TO
€
3 max{1+ f(xo).
2) — 4₁, 2}}
L,
8
xo +
1}
<max {1+ f(xo) — L\,2}.
Explain why there is a partition P₁ of [a, xo] such that U (f, P₁) — L (ƒ, P₁) < §.
- Explain why there is a partition P₂ of [xo+ ,b] such that U (ƒ, P₂) — L (ƒ, P₂) < §.
Consider the partition P = P₁UP₂ of [a, b], show that U (f, P) – L (ƒ, P) < €.
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