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has a singularity at <math>x = 0</math>, where it seems to "explode" to <math>\pm\infty</math> and is hence not defined. The absolute value function <math>g(x) = |x|</math> also has a singularity at , since it is not differentiable there.
has a singularity at <math>x = 0</math>, where it seems to "explode" to <math>\pm\infty</math> and is hence not defined. The absolute value function <math>g(x) = |x|</math> also has a singularity at , since it is not differentiable there.
在x=0处有一个<font color="#ff8000">奇点</font>，在这里它似乎“爆炸”到±∞，因此没有定义。绝对值函数g（x）=| x |在x=0处也有<font color="#ff8000">奇点</font>，因为它在那里不可微。<ref>{{cite book |first=Geoffrey C. |last=Berresford |first2=Andrew M. |last2=Rockett |title=Applied Calculus |location= |publisher=Cengage Learning |year=2015 |isbn= 978-1-305-46505-3|page=151 |url=https://books.google.com/books?id=wzNBBAAAQBAJ&pg=PA151 }}</ref>
在x=0处有一个奇点，在这里它似乎“爆炸”到±∞，因此没有定义。绝对值函数g（x）=| x |在x=0处也有奇点，因为它在那里不可微。<ref>{{cite book |first=Geoffrey C. |last=Berresford |first2=Andrew M. |last2=Rockett |title=Applied Calculus |location= |publisher=Cengage Learning |year=2015 |isbn= 978-1-305-46505-3|page=151 |url=https://books.google.com/books?id=wzNBBAAAQBAJ&pg=PA151 }}</ref>
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<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
在（x，y）坐标系中由{（x，y）：y3−x2=0}定义的代数曲线在（0,0）处有一个奇点（称为尖点）。关于代数几何中的奇点，参见代数簇中的奇点。关于微分几何中的奇点，参见奇点理论
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 I, which has two subtypes, and type 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 I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).
在实际分析中，奇点要么是不连续的，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：I型，有两种亚型；II型，也可分为两种亚型（尽管通常不是）。
* A '''type 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 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:
当f（c−）和f（c+）同时存在且为有限时，即出现第一类不连续，但是也至少适用以下三个条件中的一个：
当f（c−）和f（c+）同时存在且为有限时，即出现I型不连续，但是也至少适用以下三个条件中的一个：
** <math>f(c^-) \neq f(c^+)</math>;
** <math>f(c^-) \neq f(c^+)</math>;
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:
I型不连续性可以进一步区分为下列亚型之一:
:* 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.
* A '''type 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 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。这有两个亚型，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。
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（本性奇点）”是从复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”。”本质奇点（本性奇点）“接近无限制，即使有效解扩展到包括<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.
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.
在实际分析中，奇点或不连续是函数本身的一个性质。任何可能存在于函数导数中的奇点都被认为是属于导数，而不是原函数。
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.
在复分析中，有几类奇点。其中包括<font color="#ff8000">孤立奇点 isolated singularities</font>、<font color="#ff8000">非孤立奇点 nonisolated singularities</font> 和<font color="#ff8000">分支点 branch points</font>。
如果存在一个定义在所有U上的全纯函数g，使得对于U \ {a}中的所有z, f(z) = g(z)，那么点a是f的一个可去奇点。函数g是函数f的连续替换。<ref>{{Cite web|url=http://mathworld.wolfram.com/Singularity.html|title=Singularity|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-12}}</ref>
如果存在一个定义在所有U上的全纯函数g，使得对于U \ {a}中的所有z, f(z) = g(z)，那么点a是f的一个可去奇点。函数g是函数f的连续替换。<ref>{{Cite web|url=http://mathworld.wolfram.com/Singularity.html|title=Singularity|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-12}}</ref>
* The point ''a'' is a [[pole (complex analysis)|pole]] or non-essential singularity of ''f'' if there exists a holomorphic function ''g'' defined on ''U'' with ''g''(''a'') nonzero, and a [[natural number]] ''n'' such that ''f''(''z'') = ''g''(''z'') / (''z'' − ''a'')<sup>''n''</sup> for all ''z'' in ''U'' \ {''a''}. The least such number ''n'' is called the ''order of the pole''. The derivative at a non-essential singularity itself has a non-essential singularity, with ''n'' increased by 1 (except if ''n'' is 0 so that the singularity is removable).
* The point ''a'' is a [[pole (complex analysis)|pole]] or non-essential singularity of ''f'' if there exists a holomorphic function ''g'' defined on ''U'' with ''g''(''a'') nonzero, and a [[natural number]] ''n'' such that ''f''(''z'') = ''g''(''z'') / (''z'' − ''a'')<sup>''n''</sup> for all ''z'' in ''U'' \ {''a''}. The least such number ''n'' is called the ''order of the pole''. The derivative at a non-essential singularity itself has a non-essential singularity, with ''n'' increased by 1 (except if ''n'' is 0 so that the singularity is removable).
如果存在定义在“U”上的全纯函数“g”，且“g”（“a”）非零，且存在一个自然数“n”，使得对所有“z”属于“U”\{“a”},“f”（“z”）=“g”（“z”）/ (“z” – “a”)n，则点“a”为[[极点（复分析）|极]]或“f”的<font color="#ff8000">非本质奇点</font>。最小的这个数“n”称为“极序”。非本质奇点处的导数本身也有一个非本质奇点，当“n”增加1时（除非“n”为0，因此<font color="#ff8000">奇点</font>可移除）。
如果存在定义在“U”上的全纯函数“g”，且“g”（“a”）非零，且存在一个自然数“n”，使得对所有“z”属于“U”\{“a”},“f”（“z”）=“g”（“z”）/ (“z” – “a”)n，则点“a”为[[极点（复分析）|极]]或“f”的非本质奇点。最小的这个数“n”称为“极序”。非本质奇点处的导数本身也有一个非本质奇点，当“n”增加1时（除非“n”为0，因此奇点可移除）。
* The point ''a'' is an [[essential singularity]] of ''f'' if it is neither a removable singularity nor a pole. The point ''a'' is an essential singularity [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.<ref name=":1" />
* The point ''a'' is an [[essential singularity]] of ''f'' if it is neither a removable singularity nor a pole. The point ''a'' is an essential singularity [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.<ref name=":1" />
如果点“a”既不是可去奇点，也不是极点，则它是“f”的非本质奇点。点“a”是非本质奇点当且仅当洛朗级数具有无穷多个负次幂。
Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:
Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:
除孤立奇点外，一个变量的复变函数还可能表现出其他奇异行为。这些称为非孤立奇点，其中有两种类型:
Branch points are generally the result of a multi-valued function, such as <math>\sqrt{z}</math> or <math>\log(z)</math>, which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which are fixed in place.
Branch points are generally the result of a multi-valued function, such as <math>\sqrt{z}</math> or <math>\log(z)</math>, which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which are fixed in place.
分支点通常是一个多值函数的结果，如z√或log(z)，它们被定义在一个特定的限定域中，这样函数就可以在该域中成为单值函数。切线是被排除在域之外的一条线或曲线，用于在不连续的函数值之间引入技术分离。当真正需要切割时，该函数在分支切割的每一边都有明显不同的值。分枝切割的形状是一个选择的问题，即使它必须连接两个不同的分支点(如log(z)的z=0和z=∞)，这两个分支点是固定的。
A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and PDEs (Partial Differential Equations) – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form <math>x^{-\alpha},</math> of which the simplest is hyperbolic growth, where the exponent is (negative) 1: <math>x^{-1}.</math> More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses <math>(t_0-t)^{-\alpha}</math> (using t for time, reversing direction to <math>-t</math> so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).
A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and PDEs (Partial Differential Equations) – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form <math>x^{-\alpha},</math> of which the simplest is hyperbolic growth, where the exponent is (negative) 1: <math>x^{-1}.</math> More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses <math>(t_0-t)^{-\alpha}</math> (using t for time, reversing direction to <math>-t</math> so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).
当一个输入变量为时间时，而一个输出变量在有限时间趋于无穷大时，就会出现<font color="#ff8000">有限时间奇点 finite-time singularity</font>。这些在运动学和偏微分方程中很重要——无穷大在物理上并不存在，但在<font color="#ff8000">奇点</font>附近的行为通常是令人感兴趣的。在数学上，最简单的<font color="#ff8000">有限时间奇点</font>是x-α形式的各种指数的幂律，其中最简单的是双曲增长，其中指数为（负）1:x−1。更准确地说，为了随着时间的推移在正时间处获得<font color="#ff8000">奇点</font>（因此输出增长到无穷大），可以使用（t0−t）−α（使用t表示时间，将方向反转为−t，以便时间增加到无穷大，并将<font color="#ff8000">奇点</font>从0向前移动到固定时间t0）。
当一个输入变量为时间时，而一个输出变量在有限时间趋于无穷大时，就会出现<font color="#ff8000">有限时间奇点 finite-time singularity</font>。这些在运动学和偏微分方程中很重要——无穷大在物理上并不存在，但在奇点附近的行为通常是令人感兴趣的。在数学上，最简单的有限时间奇点是x-α形式的各种指数的幂律，其中最简单的是双曲增长，其中指数为（负）1:x−1。更准确地说，为了随着时间的推移在正时间处获得奇点（因此输出增长到无穷大），可以使用（t0−t）−α（使用t表示时间，将方向反转为−t，以便时间增加到无穷大，并将奇点从0向前移动到固定时间t0）。
An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy).
An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy).
一个例子是一个非弹性球在平面上的反弹运动。如果考虑理想化的运动，即每次弹跳动能损失的比例相同，反弹的频率就变得无限大，因为球在有限时间内静止。有限时间奇点的其他例子包括潘列夫悖论的各种形式（例如，在黑板上拖动粉笔时，粉笔会跳跃的趋势），以及在平面上旋转的硬币的进动率如何在突然停止之前加速到无限大（正如使用欧拉圆盘玩具所研究的那样）。
In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation − x = 0 }} defines a curve that has a cusp at the origin . One could define the -axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the -axis is a "double tangent."
In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation − x = 0 }} defines a curve that has a cusp at the origin . One could define the -axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the -axis is a "double tangent."
在代数几何中，代数簇的奇点是簇中切线空间可能没有规则定义的一点。奇点最简单的例子就是它们自己交叉的曲线。但是还有其他类型的奇点，比如尖点。例如，方程 -x = 0定义了一条在原点有一个尖点的曲线。可以将-轴定义为这一点的切线，但这个定义不能与其他点的定义相同。实际上，在这种情况下,-轴是一个“双切线”。
For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.
For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.
对于仿射簇和射影簇，奇点是指<font color="#ff8000"> 雅可比矩阵 Jacobian matrix</font>的秩低于簇中其他点的秩的点。
An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.
An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.
可以给出一个关于交换代数的等价定义，它扩展到抽象的簇和[[方案]]: 如果局部环在这一点上不是一个正则局部环，那么该点为<font color="#ff8000">奇点</font>。
可以给出一个关于交换代数的等价定义，它扩展到抽象的簇和[[方案]]: 如果局部环在这一点上不是一个正则局部环，那么该点为奇点。