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a. The norm ||·||: Rm→R is a continuous function. O b. Every polynomial is continuous. c. If f: Rm Rn is continuous at x in Rm and (xk) in Rm is a sequence with limk→∞Xk=X, then f(x)=limk→∞of(xk).

a. The norm ||·||: Rm→R is a continuous function. O b. Every polynomial is continuous. c. If f: Rm Rn is continuous at x in Rm and (xk) in Rm is a sequence with limk→∞Xk=X, then f(x)=limk→∞of(xk).

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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Which of the following statements are true?

 

a. The norm ||·||: R™→R is a continuous function. b. Every polynomial is continuous. c. If f: Rm→R" is continuous at x in Rm and (xk) in Rm is a sequence with limk→∞Xk=X, then f(x)=limk→∞of(xk). d. A function f: Rm→R" is continuous if and only if for every x in Rm and ε>0, there exists 8>0 such that ||f(x)-f(y)||≤ɛ holds for all y with ||x-y||≤6. e. If f: Rm→R" is a function, if x in Rm, and if f(x)=limk→∞f(xê) holds whenever (xê) in Rm is a sequence with limk→∞oXk=X, then f is continuous at x.
Transcribed Image Text:a. The norm ||·||: R™→R is a continuous function. b. Every polynomial is continuous. c. If f: Rm→R" is continuous at x in Rm and (xk) in Rm is a sequence with limk→∞Xk=X, then f(x)=limk→∞of(xk). d. A function f: Rm→R" is continuous if and only if for every x in Rm and ε>0, there exists 8>0 such that ||f(x)-f(y)||≤ɛ holds for all y with ||x-y||≤6. e. If f: Rm→R" is a function, if x in Rm, and if f(x)=limk→∞f(xê) holds whenever (xê) in Rm is a sequence with limk→∞oXk=X, then f is continuous at x.
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