ebesque measure space. Let f: [-1, 1] → R uch that there is c> 0 where for all ‚ y € [−1, 1], we have |ƒ(x) — ƒ(y)| ≤ c|x − y| - Show that f: [-1, 1] → R is continuous (and pnco measurabla)
ebesque measure space. Let f: [-1, 1] → R uch that there is c> 0 where for all ‚ y € [−1, 1], we have |ƒ(x) — ƒ(y)| ≤ c|x − y| - Show that f: [-1, 1] → R is continuous (and pnco measurabla)
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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Question
![Real Analysis
Lebesque measure space. Let f: [-1, 1] → R
such that there is c ≥ 0 where for all
x, y ≤ [−1, 1], we have |ƒ(x) — ƒ(y)| ≤ c|x − y|
E
a. Show that ƒ : [−1, 1] → R is continuous (and
hence measurable).
b. Show that If A C[-1, 1] such that mA = 0,
then mf(A) = 0.
c. Show that for every € > 0, there is a natural
number N such that if n ≥ N, then
m(ƒ ( [= ²¹², ²½ / ])) < E.
N](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6be3d30-3d01-4d2a-b677-f2dab1cd2761%2Fffcc1aa1-64ca-482d-bed8-9c84e87d4b47%2Fwdwayae_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Real Analysis
Lebesque measure space. Let f: [-1, 1] → R
such that there is c ≥ 0 where for all
x, y ≤ [−1, 1], we have |ƒ(x) — ƒ(y)| ≤ c|x − y|
E
a. Show that ƒ : [−1, 1] → R is continuous (and
hence measurable).
b. Show that If A C[-1, 1] such that mA = 0,
then mf(A) = 0.
c. Show that for every € > 0, there is a natural
number N such that if n ≥ N, then
m(ƒ ( [= ²¹², ²½ / ])) < E.
N
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