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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. |
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− | 有一个奇点在 < math > x = 0 </math > ,它似乎“爆炸”到 < math > pm infty </math > ,因此没有定义。绝对值函数 < math > g (x) = | x | </math > 也有一个奇点,因为它在那里是不可微的。
| + | 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。 |
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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{(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 |
− | 代数曲线定义于 math > (x,y) : y ^ 3-x ^ 2 = 0} </math > (x,y) </math > 坐标系在 math > (0,0) </math > 上有一个奇点(叫做尖点)。关于代数几何的奇点,请参阅《非奇异。关于微分几何的奇点,请参阅《奇点理论。
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| ==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). |
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| 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). |
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− | 在实际分析中,奇点要么是导数的不连续性,要么是导数的不连续性(有时也是高阶导数的不连续性)。有四种不连续性: i 型(有两个子类型)和 II 型(也可以分为两个子类型(虽然通常不是)。
| + | 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)。 |
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| 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: |
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− | 为了描述这两种类型的限制被使用的方式,假设 < math > f (x) </math > 是一个实参数的函数 < math > x </math > ,并且对于它的参数的任何值,比如 < math > c </math > ,那么左手限制 < math > f (c ^ -) </math > ,和右手限制 < math > f (c ^ +) </math,定义为:
| + | 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: |
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| :<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 |
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| 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> . |
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− | 值 < math > f (c ^ -) </math > 是函数 < math > f (x) </math > 趋向于下面的值 < math > x </math > 接近 < math > c </math > ,值 < math > f (c ^ +) </math > 是函数 < math > f (x) </math > 趋向于上面的值 < math > x </math > 接近 math > c </math > ,而不管函数在数学上的实际值 < 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 |
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| 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. |
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− | 不倾向于做任何事情,因为。在这种情况下,限制不是无限的,而是未定义的: 没有一个值是 < math > g (x) </math > 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,这有时被称为本质奇点。 |
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| 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. |
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− | 参数在给定值 < math > c </math > 下的可能情况如下。
| + | 参数在给定值c下的可能情况如下。 |
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| * 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>. |