(d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii) compactness, (iii) the Intermediate Value Theorem.
(d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii) compactness, (iii) the Intermediate Value Theorem.
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
![(d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by
(i) connectedness,
(ii) compactness,
(iii) the Intermediate Value Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60abd4de-ba2d-4a53-b8fc-e1d2e0799cec%2F54f484de-d263-48db-aec2-94769f0607a6%2Fwt8v9e9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by
(i) connectedness,
(ii) compactness,
(iii) the Intermediate Value Theorem.
![Consider the function f : [-1, √10] → R defined by
x²,
x,
f(x)
10
8
6
4
2
0
-1
f(x) =
0
1
x > 2;
x≤ 2.
I
2
3
4
Answer each of the following questions on the discontinuity of f.
(a) Compute and compare lim+2+ f(x) and limx→2- f(x).
(b) By using the e-8 definition, explain why limx→2 f(x) does not exist.
(c) Consider the set (0,5). Find the preimage f¹((0,5)) and use the topological
characterization of continuous functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60abd4de-ba2d-4a53-b8fc-e1d2e0799cec%2F54f484de-d263-48db-aec2-94769f0607a6%2Ffdvju6b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the function f : [-1, √10] → R defined by
x²,
x,
f(x)
10
8
6
4
2
0
-1
f(x) =
0
1
x > 2;
x≤ 2.
I
2
3
4
Answer each of the following questions on the discontinuity of f.
(a) Compute and compare lim+2+ f(x) and limx→2- f(x).
(b) By using the e-8 definition, explain why limx→2 f(x) does not exist.
(c) Consider the set (0,5). Find the preimage f¹((0,5)) and use the topological
characterization of continuous functions.
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