Assume that for any non-empty subset U CR, if U is ajar, then f¹(U) is ajar. Prove that f is continuous everywhere. Hint: Let a € R and > 0 and consider the ajar set (f(a)- e, f(a) + c) (you may assume this set is ajar without proof). You may also prove and use these facts 1) for all non-empty BCR, f(f¹(B)) CB; 2) f(A₁) ≤ f(A₂) if non-empty sets A₁, A2 satisfies A₁ CA₂ CR. Both proofs should be very short; and 3) the results of part c).
Assume that for any non-empty subset U CR, if U is ajar, then f¹(U) is ajar. Prove that f is continuous everywhere. Hint: Let a € R and > 0 and consider the ajar set (f(a)- e, f(a) + c) (you may assume this set is ajar without proof). You may also prove and use these facts 1) for all non-empty BCR, f(f¹(B)) CB; 2) f(A₁) ≤ f(A₂) if non-empty sets A₁, A2 satisfies A₁ CA₂ CR. Both proofs should be very short; and 3) the results of part c).
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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Please answer part e of this question, thank you
Transcribed Image Text:Assume that for any non-empty subset UCR, if U is ajar, then f¹(U) is ajar. Prove that f is
continuous everywhere.
Hint: Let a € R and > 0 and consider the ajar set (f(a)- e, f(a) + c) (you may assume this
set is ajar without proof). You may also prove and use these facts 1) for all non-empty BCR,
f(f¹(B)) ≤ B; 2) f(A₁) ≤ f(A₂) if non-empty sets A₁, A₂ satisfies A₁ A₂ CR. Both proofs
should be very short; and 3) the results of part c).
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