3. Let ƒ : [0, 1] → R with f(x) = 0 if x is rational and f(x) = −4 if x is irrational. Prove whether or not f is integrable.
3. Let ƒ : [0, 1] → R with f(x) = 0 if x is rational and f(x) = −4 if x is irrational. Prove whether or not f is integrable.
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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![1. Let f : [1,4] → R be defined by f(x) = 4x² for 0 ≤ x ≤ 2 and f(x) = 2x for
2 < x≤ 4. Let P = {-1,0, 2, 3, 4}. Find L(f, P) and U (f, P).
2. Let f: [0, 1] → R be defined by f(x) = -8. Prove, directly from the definition, that
f is integrable.
3. Let f: [0, 1] → R with f(x) = 0 if x is rational and f(x) = -4 if x is irrational. Prove
whether or not f is integrable.
4. Let f: [a, b] → R be integrable and let c> 0. Use the definition OR the Archimedes-
Riemann Theorem to prove directly that cf is integrable and that
-b
["(of)
" (ef) = c f². f
a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d6d9a-c2bc-4670-b756-591f350e4f00%2F13400a24-425d-4ee9-8684-f07854d36f40%2Fw4gkcaf_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let f : [1,4] → R be defined by f(x) = 4x² for 0 ≤ x ≤ 2 and f(x) = 2x for
2 < x≤ 4. Let P = {-1,0, 2, 3, 4}. Find L(f, P) and U (f, P).
2. Let f: [0, 1] → R be defined by f(x) = -8. Prove, directly from the definition, that
f is integrable.
3. Let f: [0, 1] → R with f(x) = 0 if x is rational and f(x) = -4 if x is irrational. Prove
whether or not f is integrable.
4. Let f: [a, b] → R be integrable and let c> 0. Use the definition OR the Archimedes-
Riemann Theorem to prove directly that cf is integrable and that
-b
["(of)
" (ef) = c f². f
a
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