This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 124 Quiz 2 Sept. 8, 2006 Name: 1. Show, using the rigorous definition of continuity, that the following piecewise function is not continuous at x = 0: g ( x ) = x  x  , x 6 = 0; , x = 0 Solution: Note that if x < 0 then x  x  = x x = 1 and if x > 0 then x  x  = x x = 1. Hence, lim x → g ( x ) = lim x → x  x  = lim x → ( 1) = 1 and lim x → + g ( x ) = lim x → + x  x  = lim x → + (1) = 1 This shows that lim x → g ( x ) does not exist, since the limit from the left does not equal the limit from the right. Now, if g were to be continuous at 0, it would be true that lim x → g ( x ) = g (0). However, since the limit on the left doesn’t even exist, this last equation has no hope of being true. We conclude that g is not continuous at 0. 2. Show mathematically that there is a number c with 0 ≤ c ≤ 1 such that f ( c ) = 0, where f ( x ) = 2 x 1 x . Solution: This is a case for the famous Intermediate Value Theorem. First note that f is continuous. Sinceis continuous....
View
Full
Document
This note was uploaded on 06/16/2008 for the course MATH 124 taught by Professor Kennedy during the Spring '08 term at Arizona.
 Spring '08
 KENNEDY
 Calculus, Continuity

Click to edit the document details