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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 |在也有奇点，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
==Real analysis==
==Real analysis==
实际分析
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 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 [[classification of discontinuities|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).
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）。有四种不连续性：I型，有两种亚型；II型，也可分为两种亚型（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：
:<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> .
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> .
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
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。
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.
参数在给定值c下的可能情况如下。
* 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>.