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(d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness, the Intermediate Value Theorem.

(d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness, the Intermediate Value Theorem.

Advanced Engineering Mathematics
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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Consider the function f : [-1, √10] → R defined by -G I, f(x) 10 8 6 4 2 compactness, 0 f(x) = 0 [x², x > 2; x ≤ 2. 1 x Answer each of the following questions on the discontinuity of f. Compute and compare lim 2+ f(x) and lim-2- f(x). 2 By using the ed definition, explain why lime-2 f(x) does not exist. 2 3 Consider the set (0,5). Find the preimage f-¹((0,5)) and use the topological characterization of continuous functions. the Intermediate Value Theorem. (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by connectedness,
Transcribed Image Text:Consider the function f : [-1, √10] → R defined by -G I, f(x) 10 8 6 4 2 compactness, 0 f(x) = 0 [x², x > 2; x ≤ 2. 1 x Answer each of the following questions on the discontinuity of f. Compute and compare lim 2+ f(x) and lim-2- f(x). 2 By using the ed definition, explain why lime-2 f(x) does not exist. 2 3 Consider the set (0,5). Find the preimage f-¹((0,5)) and use the topological characterization of continuous functions. the Intermediate Value Theorem. (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by connectedness,
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