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.

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 L, x₀ 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.)

http://en.wikipedia.org/wiki/Discontinuity_(mathematics)

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.