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I Let f: RR be the function defined by f(x) || if x = 0, and f(0) = a. Is there a choice of a so that f is continuous at 0? If so, what is it? If not, prove that there is not. =

I Let f: RR be the function defined by f(x) || if x = 0, and f(0) = a. Is there a choice of a so that f is continuous at 0? If so, what is it? If not, prove that there is not. =

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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Characterizations of Continuity Let f: A → R, and let CEA. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e > 0, there exists a d>0 such that |x-c| < 8 (and x € A) implies |f(x) = f(c) < €; (ii) For all Ve(f(c)), there exists a Vs (c) with the property that x Vs (c) (and xEA) implies f(x) = V(f(c)); (iii) For all (n) →c (with xn EA), it follows that f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). x-C
Transcribed Image Text:Characterizations of Continuity Let f: A → R, and let CEA. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e > 0, there exists a d>0 such that |x-c| < 8 (and x € A) implies |f(x) = f(c) < €; (ii) For all Ve(f(c)), there exists a Vs (c) with the property that x Vs (c) (and xEA) implies f(x) = V(f(c)); (iii) For all (n) →c (with xn EA), it follows that f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). x-C
= if x # 0, and Let f RR be the function defined by f(x) x f(0) = = a. Is there a choice of a so that f is continuous at 0? If so, what is it? If not, prove that there is not.
Transcribed Image Text:= if x # 0, and Let f RR be the function defined by f(x) x f(0) = = a. Is there a choice of a so that f is continuous at 0? If so, what is it? If not, prove that there is not.
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