a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R. Prove that f is integrable.
a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R. Prove that f is integrable.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 32E
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![a) Let fn: [a, b]
→ R be a sequence of integrable functions, converging uniformly to f: [a, b] → R.
Prove that f is integrable.
Hint: Show that for every partition P,
|L(f, P) – L(fn, P)| < ||f – fn|| (b – a),
and similarly for the upper sum. Combine this with Riemann's criterion.
b) Give an example of a sequence g„: [0, 1] → R of integrable functions converging pointwise to the
non-integrable function (Dirichlet's function)
g(x) = { O
1
:x irrational
:x rational](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F932dec80-8015-4c31-bdcf-744f85b7afd7%2F5fc08278-0b25-4ba4-b15b-5cc9c723943e%2F9uvt5aa_processed.png&w=3840&q=75)
Transcribed Image Text:a) Let fn: [a, b]
→ R be a sequence of integrable functions, converging uniformly to f: [a, b] → R.
Prove that f is integrable.
Hint: Show that for every partition P,
|L(f, P) – L(fn, P)| < ||f – fn|| (b – a),
and similarly for the upper sum. Combine this with Riemann's criterion.
b) Give an example of a sequence g„: [0, 1] → R of integrable functions converging pointwise to the
non-integrable function (Dirichlet's function)
g(x) = { O
1
:x irrational
:x rational
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