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b) Let f :R R , then prove that the following three statements are equivalent. (i) f is continuous at 1, (ii) Given &>0 there exists &>0 such that (x, - 8.x, + 8)Cf(f(x)-E, f(x) + ɛ) (iii) For every sequence (x,) in R such that (x)→ x, we have (f(x))→ f(x,).

b) Let f :R R , then prove that the following three statements are equivalent. (i) f is continuous at 1, (ii) Given &>0 there exists &>0 such that (x, - 8.x, + 8)Cf(f(x)-E, f(x) + ɛ) (iii) For every sequence (x,) in R such that (x)→ x, we have (f(x))→ f(x,).

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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b) Let f:R → R, then prove that the following three statements are equivalent. (i) f is continuous at *, (ii) Given & >0 there exists & >0 such that (x, - 8.x, + 8)C )-E, f(x,) + ɛ) (iii) For every sequence (x,) in R such that (x,) x, we have (f(x, ))→ f(x,).
Transcribed Image Text:b) Let f:R → R, then prove that the following three statements are equivalent. (i) f is continuous at *, (ii) Given & >0 there exists & >0 such that (x, - 8.x, + 8)C )-E, f(x,) + ɛ) (iii) For every sequence (x,) in R such that (x,) x, we have (f(x, ))→ f(x,).
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