Recall that we have two definitions for integrability. • Riemann's definition: f is integrable on [a, b] if and only if for some I e R, for any ɛ > 0, there is 8 > 0 such that for any partition P : a = xo < x1 < • · ·<< xn = b with |P| = max x; – xi–1 < 8, any choices of i EF(&)(#i – ai_1) – 1 0 L(f) = lim L(f, P). |P|→0 Hint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's definition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums. Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!
Recall that we have two definitions for integrability. • Riemann's definition: f is integrable on [a, b] if and only if for some I e R, for any ɛ > 0, there is 8 > 0 such that for any partition P : a = xo < x1 < • · ·<< xn = b with |P| = max x; – xi–1 < 8, any choices of i EF(&)(#i – ai_1) – 1 0 L(f) = lim L(f, P). |P|→0 Hint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's definition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums. Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!
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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![Recall that we have two definitions for integrability.
• Riemann's definition: f is integrable on [a, b] if and only if for some I E R, for any ɛ > 0, there is & > 0 such
that for any partition P : a = xo < x1 < • .< xn = b with P= max r; – xi–1 < 8, any choices of
i
&i E [Ti, ti-1],
- Xi-1
E.
i=1
• Darboux's definition: f is integrable on [a, b| if and only if f is bounded and the upper integral equals to lower
integral, i.e.,
U(f) = infU(f, P) = L(f)
sup L(f, P).
P
Show that these two definitions are equivalent using the Darboux's Theorem:
U(f) = lim U(f, P),
|P|¬0
L(f) = lim L(f, P).
|P|→0
Hint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's
definition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums.
Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5c3ce0c-49d4-42bd-91ad-03e86be147cf%2F1fbe7c05-2779-4ccb-bbcd-2990f99893f5%2Fwauh1cq_processed.png&w=3840&q=75)
Transcribed Image Text:Recall that we have two definitions for integrability.
• Riemann's definition: f is integrable on [a, b] if and only if for some I E R, for any ɛ > 0, there is & > 0 such
that for any partition P : a = xo < x1 < • .< xn = b with P= max r; – xi–1 < 8, any choices of
i
&i E [Ti, ti-1],
- Xi-1
E.
i=1
• Darboux's definition: f is integrable on [a, b| if and only if f is bounded and the upper integral equals to lower
integral, i.e.,
U(f) = infU(f, P) = L(f)
sup L(f, P).
P
Show that these two definitions are equivalent using the Darboux's Theorem:
U(f) = lim U(f, P),
|P|¬0
L(f) = lim L(f, P).
|P|→0
Hint: from Darboux's definition to Riemann's, it is basically an application of the squeeze theorem; from Riemann's
definition to Darboux's, use the definition of inf and sup to approximate any lower/upper sums by Riemann sums.
Be careful that you cannot assume f is continuous, so the suprema and infima may not be achieved!
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