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第47行： − 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）。有四种不连续性：I型，有两种亚型；II型，也可分为两种亚型（尽管通常不是）。+ 第97行：
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In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as the lack of differentiability or analyticity.
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as the lack of differentiability or analyticity.
在数学中，<font color="#ff8000">奇点</font>一般是一个给定数学对象没有定义的点，或一个数学对象在某些特定方面不再表现良好的点，例如缺乏可微性或可分析性。
在数学中，<font color="#ff8000">奇点 singularity</font>一般是一个给定数学对象没有定义的点，或一个数学对象在某些特定方面不再表现良好的点，例如缺乏可微性或可分析性。
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 singularity</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处也有奇点，因为它在那里不可微。
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。
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{（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）。代数奇点的多样性，参见代数几何中的奇异点。关于微分几何中的奇点，见奇点理论
在（x，y）坐标系中{（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）。代数<font color="#ff8000">奇点</font>的多样性，参见代数几何中的奇异点。关于微分几何中的<font color="#ff8000">奇点</font>，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，<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）。
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，这有时被称为本质奇点。
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>。