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Unformatted text preview: CONTINUITY We have seen that the limit of a function as x approaches a can sometimes be found by calculating the value of the function at x = a . Functions with this property are called continuous at a . Mathematical definition is as follows. Continuity A function f is continuous at a if lim x → a f ( x ) = f ( a ) . In fact, this definition requires three conditions of continuity. Those are 1. Existence of f ( a ), 2. Existence of the limit lim x → a f ( x ), and 3. lim x → a f ( x ) = f ( a ) . If any one of the above is not fulfilled, we say that f is discontinuous at a . f is continuous on an interval if it is continuous at every point on the interval. We can also define continuity from the left(right) by using the left(right)hand limit. Example 1. Find the points at which each of the following functions are dis continuous. (a) f ( x ) = x 3 1 x 1 (b) f ( x ) = H ( x ) (c) f ( x ) = ( x 2 + x + 1 if x 6 = 1 2 if x = 1 . Solution In (a), f (1) does not exist, thus it is discontinuous at 1....
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross
 Continuity

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