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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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