Editing Singularity (mathematics) (section)

==Real analysis==
In [[real analysis]], singularities are either [[classification of discontinuities|discontinuities]], or discontinuities of the [[derivative]] (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: '''type&nbsp;I''', which has two subtypes, and '''type&nbsp;II''', which can also be divided into two subtypes (though usually is not).

To describe the way these two types of limits are being used, suppose that <math>f(x)</math> is a function of a real argument <math>x</math>, and for any value of its argument, say <math>c</math>, then the '''left-handed limit''', <math>f(c^-)</math>, and the '''right-handed limit''', <math>f(c^+)</math>, are defined by:
:<math>f(c^-) = \lim_{x \to c}f(x)</math>, constrained by <math>x < c</math> and

:<math>f(c^+) = \lim_{x \to c}f(x)</math>, constrained by <math>x > c</math>.

The value <math>f(c^-)</math> is the value that the function <math>f(x)</math> tends towards as the value <math>x</math> approaches <math>c</math> from ''below'', and the value <math>f(c^+)</math> is the value that the function <math>f(x)</math> tends towards as the value <math>x</math> approaches <math>c</math> from ''above'', regardless of the actual value the function has at the point where <math>x = c</math>&nbsp;.

There are some functions for which these limits do not exist at all. For example, the function
:<math>g(x) = \sin\left(\frac{1}{x}\right)</math>
does not tend towards anything as <math>x</math> approaches <math>c = 0</math>. The limits in this case are not infinite, but rather [[Undefined (mathematics)|undefined]]: there is no value that <math>g(x)</math> settles in on. Borrowing from complex analysis, this is sometimes called an ''[[essential singularity]]''.

The possible cases at a given value <math>c</math> for the argument are as follows.
* A '''point of continuity''' is a value of <math>c</math> for which <math>f(c^-) = f(c) = f(c^+)</math>, as one expects for a smooth function. All the values must be finite. If <math>c</math> is not a point of continuity, then a discontinuity occurs at <math>c</math>.
* A '''type&nbsp;I''' discontinuity occurs when both <math>f(c^-)</math> and <math>f(c^+)</math> exist and are finite, but at least one of the following three conditions also applies:
** <math>f(c^-) \neq f(c^+)</math>;
** <math>f(x)</math> is not defined for the case of <math>x = c</math>; or
** <math>f(c)</math> has a defined value, which, however, does not match the value of the two limits.
*:
*:Type I discontinuities can be further distinguished as being one of the following subtypes:
** A '''[[jump discontinuity]]''' occurs when <math>f(c^-) \neq f(c^+)</math>, regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined.
** A '''[[removable singularity|removable discontinuity]]''' occurs when <math>f(c^-) = f(c^+)</math>, also regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
* A '''type&nbsp;II''' discontinuity occurs when either <math>f(c^-)</math> or <math>f(c^+)</math> does not exist (possibly both). This has two subtypes, which are usually not considered separately:
** An '''infinite discontinuity''' is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its [[graph of a function|graph]] has a [[vertical asymptote]].
** An '''essential singularity''' is a term borrowed from complex analysis (see below). This is the case when either one or the other limits <math>f(c^-)</math> or <math>f(c^+)</math> does not exist, but not because it is an ''infinite discontinuity''. ''Essential singularities'' approach no limit, not even if valid answers are extended to include <math>\pm\infty</math>.

In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.

===Coordinate singularities===
{{Main|Coordinate singularity}}
A '''coordinate singularity''' occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in [[spherical coordinates]]. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees).  This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles.  A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an [[n-vector|{{mvar|n}}-vector]] representation).