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Example:Points of Discontinuity

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Classification of discontinuities


Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.

This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.


Classification of discontinuities

Consider a real valued function ƒ of a real variable x, defined in a neighborhood of the point x0 in which ƒ is discontinuous. Then three situations may be distinguished:

1.   The one-sided limit from the negative direction

and the one-sided limit from the positive direction

at x0 exist, are finite, and are equal to L = L  = L + . Then, if ƒ(x0) is not equal

to Lx0 is called a removable discontinuity. This discontinuity can be 'removed

to make ƒ continuous at x0', or more precisely, the function

is continuous at x=x0.

(Our solved example in mathguru.com uses this concept)

2.   The limits L  and L + exist and are finite, but not equal. Then, x0 is called a jump discontinuity or step discontinuity. For this type of discontinuity, the function ƒ may have any value in x0.

3.   One or both of the limits L  and L + does not exist or is infinite. Then, x0 is called an essential discontinuity, or infinite discontinuity. (This is distinct from the term essential singularity which is used when studying functions of complex variables.)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.