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:<math> f(x)=\frac{1}{x} </math>
:<math> f(x)=\frac{1}{x} </math>
<math> f(x)=\frac{1}{x} </math>
<math> f(x)=\frac{1}{x} </math>
= frac {1}{ x } </math >
= frac {1}{ x } </math >
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，<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）。
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）。
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.
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。借用复分析，这有时被称为<font color=“#ff8000”>本质奇点（本性奇点） essential singularity </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。借用复分析，这有时被称为<font color="#ff8000">本质奇点（本性奇点） essential singularity</font>。
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>.
“<font color=“#ff8000”>本质奇点</font>”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”。<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>。
“ <font color="#ff8000">本质奇点（本性奇点）</font>”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”。<font color="#ff8000">”本质奇点（本性奇点）“</font>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，<font color="#ff8000">奇点</font>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<font color="#ff8000">奇点</font>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<font color="#ff8000">奇点</font>。其中包括孤立奇点、非孤立奇点和分支点。
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 non-essential singularity</font>。最小的这个数“n”称为“极序”。 <font color=“#ff8000”>非本质奇点</font>处的导数本身也有一个<font color=“#ff8000”>非本质奇点</font>，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<font color=“#ff8000”>非本质奇点 non-essential singularity</font>。最小的这个数“n”称为“极序”。 <font color=“#ff8000”>非本质奇点</font>处的导数本身也有一个<font color=“#ff8000”>非本质奇点</font>，当“n”增加1时（除非“n”为0，因此<font color="#ff8000">奇点</font>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 <font color=“#ff8000”>非本质奇点</font>。点“a”是 <font color=“#ff8000”>非本质奇点</font>[[iff |当且仅当][[Laurent级数]]具有无穷多个负次幂。
如果点“a”既不是可去奇点，也不是极点，则它是“f”的 <font color=“#ff8000”>非本质奇点</font>。点“a”是 <font color=“#ff8000”>非本质奇点</font>[[iff |当且仅当][[Laurent级数]]具有无穷多个负次幂。