We know that is continuous at if and only if . Thus, from the properties of the one-sided limits, which we denote by and , it follows that is discontinuous at if and only if , or .
If , but , then is called a removable discontinuity of . If we modify the value of at to , then will become continuous at . This is clear since the modified (call it ) satisfies
If , but is not in (so is not defined), then is also called a removable discontinuity. If we assign , then this modification renders continuous at .
If , then has a jump discontinuity at Then the number is then called the jump, or saltus, of at .
If either (or both) of or does not exist, then has an essential discontinuity at (or a discontinuity of the second kind).
Note that may be continuous (continuous in all points in ), but still have discontinuities in
Consider the function given by
Since , , and , it follows that has a removable discontinuity at . If we modify so that , then becomes the continuous function .
Let us consider the function defined by the formula
where is a nonzero real number. When , the formula is undefined, so is only determined for . Let us show that this point is a removable discontinuity. Indeed, it is easy to see that is continuous for all , and using L’Hôpital’s rule (http://planetmath.org/LHpitalsRule) we have . Thus, if we assign , then becomes a continuous function defined for all real . In fact, can be made into an analytic function on the whole complex plane.
The signum function is defined as
Since , , and since , it follows that has a jump discontinuity at with jump .
The function given by
has an essential discontinuity at . See  for details.
Let be topological spaces, and let be a mapping . Then is discontinuous at , if is not continuous at (http://planetmath.org/Continuous) .
In this generality, one generally does not classify discontinuities quite so closely, since they can have quite complicated behaviour.
A jump discontinuity is also called a simple discontinuity, or a discontinuity of the first kind. An essential discontinuity is also called a discontinuity of the second kind.
- 1 R.F. Hoskins, Generalised functions, Ellis Horwood Series: Mathematics and its applications, John Wiley & Sons, 1979.
- 2 P. B. Laval, http://science.kennesaw.edu/ plaval/spring2003/m4400_02/Math4400/contwork.pdfhttp://science.kennesaw.edu/ plaval/spring2003/m4400_02/Math4400/contwork.pdf.
|Date of creation||2013-03-22 13:45:01|
|Last modified on||2013-03-22 13:45:01|
|Last modified by||mathwizard (128)|
|Defines||discontinuity of the second kind|
|Defines||discontinuity of the first kind|