# Lect10 - Continuity Definition point x = a If Let f(x be a...

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92.131 Lecture 10 1 of 10 Ronald Brent © 2009 All rights reserved. Continuity Definition: Let ) ( x f be a function defined on an interval I containing the point a x = . If ) ( ) ( lim a f x f a x = then ) ( x f is continuous at a x = . A function that is NOT continuous at a point a x = is said to be discontinuous there. If ) ( x f is continuous at every point in an interval I , then ) ( x f is said to be continuous on I . Note: The definition for continuity really has three parts: 1. The function must be defined at a . 2. The limit should exist as x approaches a . 3. The definition and limit must be equal.

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92.131 Lecture 10 2 of 10 Ronald Brent © 2009 All rights reserved. Example: Where is the function 1 1 ) ( 2 = x x x f continuous? We examine the 3 requirements: 1. Since function is not defined at 1 = x , it can’t be continuous there. At every other point 1 a , 1 ) 1 ( ) 1 )( 1 ( 1 1 ) ( 2 + = + = = a a a a a a a f 2. The limit ) ( lim x f a x exists everywhere, (even for 1 = x ,) and 1 ) 1 ( lim 1 1 lim ) ( lim 2 + = + = = a x x x x f a x a x a x . 3. The limit equals the definition at every point 1 = a x , so the function is continuous for 1 x .

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Lect10 - Continuity Definition point x = a If Let f(x be a...

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