Another type is an essential discontinuity. For example, take the function f(x) = sin (1/x). The function is continuous everywhere except at x = 0. However, the limit as x approaches 0 of this function does not exist (the value of the function oscillates ever more quickly between 1 and -1), so there is no way to repair the function at 0 to make it continuous. In this case, we say sin (1/x) has an essential discontinuity at x = 0.
Formally: We say f has an essential discontinuity at a point x_0 when either the limit of f(x) as x approaches x_0 from the left does not exist, or the limit of f(x) as x approaches x_0 from the right does not exist.
A third type is a jump discontinuities. For example, consider the function
f(x) =
0, if x < 0
1, if x >= 0.
The function is not continuous at x = 0, because there is a gap in the graph there. Clearly this discontinuity cannot be "fixed" by changing the value of f(x) at x = 0, so it isn't a removable discontinuity. However, if we choose f(0) = 1, then we can at least make the graph continuous from the right at 0. (We could also define f(0) = 0 to make it continuous from the left.) A graph which is continuous on one side only in this manner is said to have a jump discontinuity.
Formally: A function f has a jump discontinuity at x_0 if the limit of f(x) as x approaches x_0 from the right exists, and the limit of f(x) as x approaches x_0 from the left exists, but the two limits are not equal.
2007-08-30 17:26:30
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answer #1
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answered by Anonymous
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Infinite, jump, oscillating.
2007-08-31 00:21:08
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answer #3
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answered by swd 6
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