(c) Let f, g, h R → R be functions such that for all a we have f(x) ≤ g(x) ≤ h(r) and f(0) = g(0)h(0). Prove that if f and h are continuous at 0 then g is continuous at 0. (d) Prove that the function f: R→ R defined by f(x) = {rsin (2) is continuous. You may assume that sin(x) is a continuous function on all of R which takes values between 1 and 1. x 40 x=0.
(c) Let f, g, h R → R be functions such that for all a we have f(x) ≤ g(x) ≤ h(r) and f(0) = g(0)h(0). Prove that if f and h are continuous at 0 then g is continuous at 0. (d) Prove that the function f: R→ R defined by f(x) = {rsin (2) is continuous. You may assume that sin(x) is a continuous function on all of R which takes values between 1 and 1. x 40 x=0.
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
part c and d
![2. CONTINUOUS FUNCTIONS
For each of these problems you may use either of our two equivalent notions of continuity:
sequences or e-8. But be aware that you will be expected to be familiar with both methods.
(a) Assume that f: [0, 0)→ R and g: (-00, 0] are continuous functions such that f(0) = g(0).
Prove that the function f: R→ R defined by
f(x)
h(x) = { g(x)
is continuous with the e-d method. Then prove it with the sequence method.
(b) Prove that there is no continuous function f: R→R such that for all 2 ER\ {0} we have
f(x) = 1.
(c) Let f, g, h R → R be functions such that for all r we have f(r) ≤ g(r) ≤ h(r) and
f(0) = g(0)h(0). Prove that if f and h are continuous at 0 then g is continuous at 0.
(d) Prove that the function f: R→IR defined by
x sin (-)
x>0
x < 0.
f(x) = { 3²
0
x‡0
x = 0.
is continuous. You may assume that sin(x) is a continuous function on all of R which takes
values between -1 and 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2F571695e4-0e48-42a7-921b-ff0f98ecb787%2Fgnkc3kb_processed.png&w=3840&q=75)
Transcribed Image Text:2. CONTINUOUS FUNCTIONS
For each of these problems you may use either of our two equivalent notions of continuity:
sequences or e-8. But be aware that you will be expected to be familiar with both methods.
(a) Assume that f: [0, 0)→ R and g: (-00, 0] are continuous functions such that f(0) = g(0).
Prove that the function f: R→ R defined by
f(x)
h(x) = { g(x)
is continuous with the e-d method. Then prove it with the sequence method.
(b) Prove that there is no continuous function f: R→R such that for all 2 ER\ {0} we have
f(x) = 1.
(c) Let f, g, h R → R be functions such that for all r we have f(r) ≤ g(r) ≤ h(r) and
f(0) = g(0)h(0). Prove that if f and h are continuous at 0 then g is continuous at 0.
(d) Prove that the function f: R→IR defined by
x sin (-)
x>0
x < 0.
f(x) = { 3²
0
x‡0
x = 0.
is continuous. You may assume that sin(x) is a continuous function on all of R which takes
values between -1 and 1.
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