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The algebraic curve defined by <math>\{(x,y):y^3-x^2=0\}</math> in the <math>(x, y)</math> coordinate system has a singularity (called a cusp) at <math>(0, 0)</math>. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.

The algebraic curve defined by <math>\{(x,y):y^3-x^2=0\}</math> in the <math>(x, y)</math> coordinate system has a singularity (called a cusp) at <math>(0, 0)</math>. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.

在（x，y）坐标系中由{（x，y）：y3−x2=0}定义的代数曲线在（0,0）处有一个<font color="#ff8000">奇点</font>（称为尖点）。关于代数几何中的<font color="#ff8000">奇点</font>，参见代数簇中的奇异点。关于微分几何中的<font color="#ff8000">奇点</font>，参见<font color="#ff8000">奇点</font>理论

在（x，y）坐标系中由{（x，y）：y3−x2=0}定义的代数曲线在（0,0）处有一个<font color="#ff8000">奇点</font>（称为尖点）。关于代数几何中的<font color="#ff8000">奇点</font>，参见代数簇中的<font color="#ff8000">奇点</font>。关于微分几何中的<font color="#ff8000">奇点</font>，参见<font color="#ff8000">奇点</font>理论

In real analysis, singularities are either 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).

In real analysis, singularities are either 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).

在实际分析中，<font color="#ff8000">奇点</font>要么是不连续的，要么是导数的不连续（有时也是高阶导数的不连续）。有四种不连续性：I型，有两种亚型；II型，也可分为两种亚型（尽管通常不是）。

在实际分析中，<font color=“#ff8000”>奇点</font>要么是不连续的，要么是导数的不连续（有时也是高阶导数的不连续）。有四种不连续：类型一，有两种子类型；类型二，也可分为两种子类型（尽管通常不是）。

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:

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:

为了描述这两种极限的使用方式，假设f（x）是实参x的函数，对于其自变量的任何值，比如c，则左极限f（c-）和右极限f（c+）的定义如下：

为了描述这两种极限的使用方式，假设<math>f(x)</math>是实参数x的函数，对于其自变量的任意值，比如c，则左极限f（c-）和右极限f（c+）的定义如下：

 

:<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> 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 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;.

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;.

值f（c-）是函数f（x）在值x从下面接近c时趋于的值，而值f（c+）是函数f（x）在值x从上接近c时趋向的值，而不管函数在x=c点处的实际值如何  

值f（c-）是函数f（x）在值x从下接近c时趋向的值，而值f（c+）是函数f（x）在值x从上接近c时趋向的值，而不管函数在x=c点处的实际值如何  

 

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: there is no value that <math>g(x)</math> settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

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: there is no value that <math>g(x)</math> settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

不倾向于做任何事情，因为。在这种情况下，极限不是无限的，而是未定义的: 没有一个值是g (x)确定的。借用复杂的分析，这有时被称为本质<font color="#ff8000">奇点</font>。

在x趋于<math>c = 0</math>时不趋向任何值。在这种情况下，极限不是无限的，而是没有定义的:g(x)m没有确定的值。借用复杂的分析，这有时被称为<font color=“#ff8000”>本质奇点（本性奇点） essential singularity </font>。

The possible cases at a given value <math>c</math> for the argument are as follows.

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 '''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:

* 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(c^-) \neq f(c^+)</math>;

** <math>f(x)</math> is not defined for the case of <math>x = c</math>; or

** <math>f(x)</math> is not defined for the case of <math>x = c</math>; or

:Type I discontinuities can be further distinguished as being one of the following subtypes:

: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 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.

* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管定义了f（c）的值，都会出现跳跃不连续

* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管是否定义了f（c）的值，都会出现跳跃不连续。

:* 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 '''[[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 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 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).

* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管是否已定义它的值(但不匹配两个极限)时，就会出现可移动的不连续性。

* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管是否已定义它的值(但不匹配两个极限)时，都会出现可移动的不连续性。

* 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:

* 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:

当f（c−）或f（c+不存在时（可能两者都不存在），就会出现“II”型不连续性。这有两个子类型，通常不单独考虑：

当f（c−）或f（c+不存在时（可能两者都不存在），就会出现“II”型不连续性。这有两个子类型，通常不单独考虑：

** 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 '''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]].

无限不连续”是左手极限或右手极限都不存在的特例，特别是因为它是无限的，而另一个极限要么是无限的，要么是某种定义良好的有限数。换句话说，当函数的[[函数的图|图]]有一个[[垂直渐近线]]时，函数具有无限的不连续性

无限不连续是当左极限或右极限不存在时的特例，特别是因为它是无限的，而另一个极限要么是无限的，要么是某种定义良好的有限数。换句话说，当函数的图形有一个[[垂直渐近线]]时，函数具有无限的不连续性。

** 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>.

** 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>.

“本质奇点”是从复杂分析中借用的术语（见下文）。当一个或另一个极限不存在时，这种情况不是因为它是一个“无限不连续性”基本奇点“接近无限制，即使有效答案扩展到包括<math>\pm\infty</math>  

“<font color=“#ff8000”>本质奇点</font>”是从复杂分析中借用的一个术语（见下文）。当极限f（c−）或f（c+）两者中的任意一者不存在时，情况就会如此，但不是因为它是一个“无限不连续性”。<font color=“#ff8000”>本质奇点</font>“接近无限制，即使有效解扩展到包括<math>\pm\infty</math>。

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 -vector representation).

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 -vector representation).

 

==Complex analysis复杂分析==

==Complex analysis复分析==

In [[complex analysis]], there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.

In [[complex analysis]], there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.

In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.

In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.