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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,