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Define f : R" → [0, 0) by letting f(x) = (1 – ||æ||<)*72 if ||x|| < 1 and f(x) = ( if ||x|| > 1. Prove that f is continuous on R™. (To prove that f is continuous at a given xo E R", consider separately the following three cases: ||xo|| > 1, ||xo|| < 1, ||x0|| = 1.)

Define f : R" → [0, 0) by letting f(x) = (1 – ||æ||<)*72 if ||x|| < 1 and f(x) = ( if ||x|| > 1. Prove that f is continuous on R™. (To prove that f is continuous at a given xo E R", consider separately the following three cases: ||xo|| > 1, ||xo|| < 1, ||x0|| = 1.)

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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Please solve this question by using the lemma for each of the 3 cases. In this case A is just R^m

A rel. to A if and and mly f Lemma 3 fiA? →RM is conti. at only V E >0, 3 foo st. Hw - fin| « £ v xE Brla) n A
Transcribed Image Text:A rel. to A if and and mly f Lemma 3 fiA? →RM is conti. at only V E >0, 3 foo st. Hw - fin| « £ v xE Brla) n A
Define f : Rm → [0, 0) by letting f(x) = (1 – ||æ||2)1/2 if ||x|| < 1 and f(x) = 0 if ||x|| > 1. Prove that f is continuous on Rm. (To prove that f is continuous at a given xo E R", consider separately the following three cases: ||xo|| > 1, ||xo|| < 1, ||xo|| = 1.)
Transcribed Image Text:Define f : Rm → [0, 0) by letting f(x) = (1 – ||æ||2)1/2 if ||x|| < 1 and f(x) = 0 if ||x|| > 1. Prove that f is continuous on Rm. (To prove that f is continuous at a given xo E R", consider separately the following three cases: ||xo|| > 1, ||xo|| < 1, ||xo|| = 1.)
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