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Unformatted text preview: Continuity A process is said to be continuous if it happens in a gradual manner, without interruptions or abrupt changes. This can be described in a simple way with limits. Definition. We say a function f is continuous at a number a if lim x → a f ( x ) = f ( a ). This says, f ( x ) is close to f ( a ) if x is close to a . So the definition means, there is no break in the graph of f at a . There are several kinds of discontinuities. Example. Consider f ( x ) = x 2 4 x 2 at x = 2. Now f (2) is undefined, so we cannot have lim x → 2 f ( x ) = f (2). So f is not continuous at x = 2. Now if x 6 = 2 then f ( x ) = x 2 4 x 2 = ( x 2)( x + 2) x 2 = x + 2. So the graph of f looks exactly like the line y = x + 2 but with the point (2 , 4) missing. So this discontinuity is sometimes referred to as a “missing point” discontinuity. More often, it is called a “removable” discontinuity; it can be removed by filling in one point in the graph of f . Put differently, if we define g ( x ) = x 2 4 x 2 if x 6 = 2 4 if x = 2 , we produce a function that is exactly the same as f except at x = 2, but which is contin uous. That is, we redefine f at exactly one point ( x = 2), and “remove” the discontinuity of f . Of course, the function g is just g ( x ) = x + 2 for all x . In general, a function f has a removable discontinuity at x = a provided lim x → a f ( x ) = f ( a ) but f ( a ) is not defined or f ( x ) 6 = f ( a ). In the example, lim x → 2 f ( x ) = 4 does exist, but f (2) does not exist (and does not equal 4).does not exist (and does not equal 4)....
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This note was uploaded on 01/19/2009 for the course MAT 1214 taught by Professor Johnrayko during the Spring '08 term at The University of Texas at San Antonio San Antonio.
 Spring '08
 JohnRayko
 Calculus, Continuity, Limits

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