A suppose f x is a function defined for all real

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(a) Suppose f ( x ) is a function defined for all real numbers, and that x 2 f ( x ) x 2 for every x . Then f ( 0 ) = 0 and f ( x ) is continuous at x = 0 . Evaluating the expression x 2 f ( x ) x 2 at x = 0 , we get 0 2 f ( 0 ) 0 2 . In particular, 0 f ( 0 ) 0 , if f ( 0 ) is defined. Since we are given that f is defined for all real numbers, we may conclude that f ( 0 ) = 0 . Further, note that lim x 0 x 2 = 0 = lim x 0 x 2 . We may conclude by the sandwich theorem that lim x 0 f ( x ) = 0 . In particular, lim x 0 f ( x ) = 0 = f ( 0 ) , so f ( x ) is continuous at x = 0 . (b) If lim x a + f ( x ) = lim x a f ( x ) , then f ( x ) is continuous at x = a . Consider the example: f ( x ) = 0 x < 0 ; 1 x = 0 ; and 0 x > 0. We have lim x 0 + f ( x ) = 0 = lim x 0 f ( x ) , and so lim x 0 f ( x ) = 0 . The function is not continuous at x = 0 , however, because f ( 0 ) = 1 .
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Homework 4 Book Problem Answers Section 2..5: Section 2.6:
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  • Fall '06
  • MARTIN,C.
  • Calculus, bad check, presentation problems

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