从“连续王国”走向“可测共和国”
---学习测度论的一些体会

【中英对照关键词表】

“连续王国”：The Realm of Continuous Functions (The Realm of Continuity)

“可测共和国”：The Republic of Measurable Functions (The Republic of Measurability)

【引言】

季候风网友在点评兄弟关于拓扑和Sigma代数之间联系的最初印象时谈到：

“可数可加” 是最关键的概念，是 “测度” 这个概念赖以生存的基础，如果不是因为这个限制，
平面里的直线就会有非零的测度，我们对于测度的原始概念（长度，面积，体积，概率...）就会被颠覆。

这段话进一步激发了兄弟在阅读Gauge兄关于均匀分布和测度论的文章时已经产生的深入了解Lebesgue测度的兴趣。我原本已经打算放下对Walter Rudin几本教科书的成见，想以他的"Real and Complex Analysis"为深入学习的入手点。令人遗憾的是此书虽然提供了一个比"Principles of Mathematical Analysis"略微友好的界面，但我在读到拓扑和Sigma代数的概念类比后的兴致刚浓，就被Rudin不断抛出的术语和定理弄得眼花缭乱，被迫正式宣告Rudin的书只适合数学系的学生。

顺便在此谈一下对数学教育的个人看法。Rudin这类权威教科书作者显然是数学界的主流，他们沉迷于在有限的著作空间里塞入最大的知识信息量，从而获得充分展示学识深度的自我满足感。在大多数学生抱怨其书难读时，他们为自己辩解的理由是读者在学习时必须全力以赴，在你的背景知识积累够量后才能体会理解的乐趣。我认为这是目前数学教育的一个主要误区，当代社会的高节奏早已和这种中古时代的治学方法无法相容，而柏杨先生在《白话资治通鉴》的序言中就已表达了与我类似的观点。这些“纯粹的”数学教育家大概以为世界上只有数学一种学问，学生在学习时就是不能求快，这就相当于要求现代中国社会对历史有兴趣的年轻人要追求国粹，去读线装竖排且没有标点的古文原版《资治通鉴》。他们鼓吹在你勤奋学习，达到能将标点补入原著并通读司马光流畅的文言文之后，你对中国文史的理解将上一个新的台阶。问题是一般读者都不是古汉语专业的，如此要求显然非常荒唐。即便像我完全可以通读文言文，但由于工作生活忙碌和兴趣爱好广泛，更愿意在最短的时间内了解《资治通鉴》的主要内容，因此选择阅读柏杨的白话版是当然的。不过这种选择的前提是我对柏杨的学识有足够的信心，一般作者即使愿意翻译也不容易在我心中确立其可信度(credibility)。

我手上的Rudin“老三篇”经典都是国内“华章图书”的正式影印版，其封底都有一段用中文写的点评，有趣的是"Principles of Mathematical Analysis"和"Real and Complex Analysis"的中文点评者的观点不太一致：

* "Principles of Mathematical Analysis"的评论者在赞扬完Rudin之后加了一句：“本书内容相当精炼，结构简单明了，这也是Rudin著作的一大特色。与其说这是一部教科书，不如说这是一部字典。”

* 而"Real and Complex Analysis"的评论者就缺乏冷静客观的头脑，只知道赞美：“毫不夸张地说，掌握了本书，对数学的将会上一个新的台阶。”

我在断然抛弃Rudin著作之余，也深感数学教育界迫切需要大量类似柏杨那样的专家肯花时间担任翻译和注释者的工作来造福青年学生和非本专业的兴趣自学者。幸运的是，欧美数学界确实有这样具有高可信度的优秀数学教育家，William Dunham教授无疑是其中的典范之一。我在2005年读了从图书馆借出的"Journey Through Genius"几个章节之后，就果断在Amazon买了Dunham教授全部四本著作--- "Journey Through Genius", "Euler---The Master of Us All", "The Mathematical Universe", 和最新的"The Calculus Gallery"。令兄弟非常欣慰的是，"The Calculus Gallery"成为我这个长周末轰开Lebesgue测度论这扇铁门的一门重炮。

另外最近在图书馆借的数学经典论文集"God Created the Integers"也派上了用场，该书主编Stephen Hawking的主要贡献是为其中17名数学家的每一位写了生平简介，其中关于Lebesgue的那篇成为学习Dunham著作时的重要辅助。此论文集的内容编排非常出色，价格也是惊人的便宜，但其两大致命弱点使我认为还不能急于收藏其2005年初版：

* 目前的版本居然没有索引(index)，而此书最大的价值恐怕就在查阅；

* 在众多数学公式的印刷过程中引入了大量的错误，这对一本非原创的选集类出版物来讲非常不应该。

只有在不久的将来这两大缺陷在第二版中得到纠正之后，此书才会真正具备典藏价值。

兄弟近来的一大乐趣是思考Bayesian统计学和逻辑概率论的本质，而要想深入理解这两个相关领域，Lebesgue测度这一核心概念显然是无法回避的。概率论已经是数学教育中公认的难点，而测度论深藏在概率论的背后，对于从未系统学习过概率论的我来讲更显神秘。

统计学和概率论其实差别挺大，前者的学习可以通过数据分析的实践来“边走边打”，这是所有实用类学科的优势。而概率论基本上还属于纯粹数学范畴，其学习离不开历史背景，这也是数学教育难于其它自然科学教育的关键。一般自然科学新的理论通常超越旧的理论，几百年前的化学专家对原子论还不太了解，而如今的小学生都知道原子的基本结构。现代的学生学习化学时只需略微知道一点本学科的历史，不了解历史对其学习并无直接影响。而检验数学理论的标准是内在的自洽，几千年前的欧氏几何依然是当代中学数学的主要内容，数学史对学习现代数学来讲非常重要。这也是我非常佩服数学家的原因，而数学在过去几百年里的发展又是极其迅猛，当代数学家在能够独立做研究之前所需要付出的在学习上的投资是非常巨大的。而我从来对数学就采取旁观者的态度，“弱水三千，只取一瓢饮”的战略让我只关注与统计学有关的基础数学。在信息爆炸的时代，我们对于许多无关全局的知识的学习有必要采取“观其大略，不求甚解”的战略，但是对于到处都是陷阱的统计学和概率论而言，很多知识点都能影响全局，因此兄弟经常会被迫花时间去求甚解。

Lebesgue测度就是概率论中可以影响全局的一个概念，要想透彻理解必须先把数学分析中的积分和可积形定义的历史演变搞清楚。Rudin名著"Real and Complex Analysis"只知填鸭式地抛出大量定义和定理及其证明，对于Lebesgue积分的历史背景非常吝惜笔墨：

"Towards the end of the 19th century it became clear to many mathematicians that the Riemann integral should be replaced by some other type of integral, more general and more flexible, better suited for dealing with limit processes. ... It was Lebesgue's construction which turned out to be the most successful."

接下来Rudin故作高深地用Riemann时代还没有的测度语言来概括Riemann积分的大意，随后又介绍了许多有用的术语和定义。虽然Rudin非常可取地点出了拓扑和Sigma代数之间的类比反映出连续函数和可测函数的联系与区别，但他并没有抓住问题的要害。在可测性这一基本概念还没讲清的情况下，又迫不及待地抛出Borel集合与简单函数这类偏离主题的新概念，如此混乱的“枪法”对非数学专业的读者来讲是无法忍受的。

Dunham教授的近著"The Calculus Gallery"读来令人赏心悦目，而我在Hawking教授评论Lebesgue生平时了解到积分定义的演变必须抓住"Cauchy-->Riemann-->Lebesgue"这条历史主线，于是就只读Dunham书中与此三人有关的章节。只用了几个小时的时间，就理出如下清晰的头绪，为了概念表达的精确将以英语表述为主：

(1) Leibniz originally defined the integral as a sum of infinitely many infinitesimal summands in the context of finding the area under a curve. But by 1800, algebraic notions started to dominate over geometrical notions in calculus under the influence of Euler's great contributions. Integral has come to be regarded merely as the inverse of differentiation, indefinite integral now has the nickname of "antiderivative". But this kind of treatment of integral severely underestimated its profound nature.

(2) Cauchy believed that the integral must have an independent existence and put definite integral back into the spotlight. He restored the usage of "limit" to define a definite integral across an interval. Although Cauchy's definition had the major weakness of limiting to continuous functions, he did make two critical points:

* The integral is a limit of a summation
* Integral's existence has nothing to do with antidifferentiation

After putting the concept of definite integral onto the solid foundation of limit, he was able to prove the fundamental theorem of calculus with enough mathematical rigor.

(3) Dirichlet's function has infinitely many discontinuities in any given interval:

D(x) = 1 if x is rational,
D(x) = 0 if x is irrational

Cauchy's integral couldn't handle the integration of this function. Dirichlet suggested that a new and more inclusive theory of integration might be crafted to handle such functions, a theory connected to "the fundamental principles of infinitesimal analysis". This challenge was taken up by Bernhard Riemann in 1854.

(4) Riemann proposed a bold and provocative idea to divorce integrability from continuity! Riemann将那些在连续函数占统治地位的王国中被”歧视“的一些函数予以解放(deliverance)，这些从”连续王国“的专制体制下获得自由的函数暂时无家可归，Riemann就给它们搭了一座名叫"Riemann Integrable Functions"的”流动难民营“。但是这些有界函数”难民“中颇有一些”亡命之徒“，在很窄的区间内取值照样剧烈振荡，很难”管教“。Riemann为了建造这所”难民营“可谓苦心孤诣，其数学细节涉及很多公式，无法在本坛交流。但是其核心思想可以表述如下：

In order for a function to have a Riemann integral, its oscillations must be under control. A function that jumps too often and too wildly cannot be integrated. From a geometrical viewpoint, such a function would seem to have no definable area under the curve.

最后这句话不幸将混入”难民队伍“的Dirichlet函数作为”不法分子“揪了出来，该函数实在是”行为偏僻性乖张“，居然连”难民营“都不能容它，呵呵。

(5) The fact that Dirichlet's function is not even Riemann integrable raised a fundamental question: how discontinuous can a function be and still be integrable by Riemann's definition? The answer to this question had to wait until the 20th century to be provided by Henri Lebesgue.

(6) In his 1902 doctoral dissertation titled "Intégrale, longueur, aire", Lebesgue provided insights that revolutionized integration theory and real analysis. He did so by returning to the concepts of length and area, viewing them from a fresh perspective, and therefore providing an alternative definition of the integral.

He started with an intuitive definition that the "length" of any of the four real intervals [a,b], (a,b]. [a,b), and (a,b) is simply b-a. The concept of "length" can be generalized to the concept of "measure". The measure of a bounded set can then be derived from the measure of an interval.

下文中的"measure"(测度)，"length"(长度)，甚至"width"(宽度)都将作为同义词互换使用。

Before Lebesgue, Alan Harnack introduced the so-called "outer content" of a bounded set. Given such a set, he began by enclosing it within a covering of finitely many intervals and using the sum of their lengths as an approximation to the set's outer content.

“用几个无重叠区间来覆盖一个集合”并用相应的区间测度相加来定义该集合的测度是一个非常重要的思想，而Harnack定义中的"finitely"是个关键词，这是他错失历史性突破良机的一个要点。如果他能想到将条件放宽到"coutably"，那么巨大的荣誉将属于他而不是Lebesgue。在此不妨将李白名句改一个字来发一下感慨：

古来圣贤皆寂寞，唯有赢者留其名。

Harnack借鉴了Riemann积分的核心思想又提出了下面这个大胆想法：

Cover a bounded set E by finitely many intervals in all possible ways, sum the lengths of the intervals in each coverings, and define the outer content of E (written as c(E)) to be the limit of such sums as the length of the widest interval goes to zero.

Using this bold idea, Harnack found that an infinite, nowhere-dense set has zero outer content. This was a very satisfactory result according to our intuition. But he ran into difficulty with the set of rational numbers (Q_1) in [0,1]: an infinite, dense set. He recognized that any covering of Q_1 by a finite number of intervals will of necessity cover all of [0,1]. Hence c(Q_1) = 1-0 = 1. Applying similar reasoning to the set of irrationals (I_1) in [0,1], he found that c(I_1) = 1 as well. Because the union of the disjoint sets Q_1 and I_1 is the entire interval [0,1], we see that

c(Q_1 ∪ I_1) = c([0,1]) =1

yet

c(Q_1) + c(I_1) = 1 + 1 = 2

So we end up with a "disjoint nonadditivity", which is a very unpleasant feature of Harnack's theory of content.

这里既然写到了"of necessity"，就顺便提一下老马《资本论》中那句让人耳熟能详的名言“从必然王国到自由王国”的英语表述，这里的关键是“王国”一词要用"realm"，而不能用"kingdom"，因为前者还有“界”，“领域”，“范围”等多层内涵 ---

必然王国：The Realm of Necessity

自由王国：The Realm of Freedom

本文的标题显然是受了这句名人名言以及著名电视剧《走向共和》的启发而得。

Lebesgue got the major historical credit by providing a final solution to this problem. He defined a set to have "outer measure" zero if it "can be enclosed in a finite or a denumerable infinitude of intervals whose total length is as small as we wish".. The innovation here is that Lebesgue, unlike Harnack, permitted coverings by a "denumerable infinitude" of intervals, and this made a world of difference.

Lebesgue允许“可数的无穷个区间”来覆盖一个集合是合理测度有理数集合的关键！根据Cantor著名的定理，任何一个区间内的有理数有可数的无穷多个，而同一区间内的无理数则有不可数的无穷多个。实数轴上单独一个数可看成一个点，它可用一个测度为0的无穷小宽度的区间来覆盖。可数的无穷多个零测度区间的总测度仍然是0，从而任何一个区间内的有理数集合的Lebesgue外测度为0，而同一区间内的无理数集合之测度由于是不可数的无穷多个无穷小相加，其和不是0，Lebesgue将其外测度定义为该区间的长度。这就巧妙解决了Harnack所面临的不可加性问题！写下这段文字让我深刻体会到用中文表达数学思路的相对不简洁，同样的意思用英语来表述读来就要流畅许多。

In other words, Lebesgue defined a set's outer measure in terms of coverings of intervals that form a superset of the set being measured. Each interval in a covering has a measure, and the sum of these measures may be finite. Since a bounded set can always be covered by the entire bounded interval, Lebesgue recognized that the sum of the measures of the coverings of a bounded set always has a finite infimum (greatest lower bound). This infimum is defined as the set's outer measure.

在此提一下Hawking在其文章中论及上述Lebesgue观点时出现的一个明显错误，先引用一下老霍的原文：

"Lebesgue demonstrated that the rationals have outer measure of 0 by employing Cantor's demonstration that the rationals can be enumerated and by enclosing the nth rational number in a closed interval of width (measure) ε^n. Summing the measures of all of these intervals results in ε which can be made as small as desired. Thus the rationals have an outer measure of 0, ..."

这里老霍的ε^n要么是印刷错误，要么是他老人家的计算错误。按照他的说法，该有理数集合的总外测度该是：

ε + ε^2 + ε^3 + ... + ε^n + ... = lim ε(1-ε^n)/(1-ε) = ε/(1-ε), which is not equal to ε (here the limit is taken as n approaches infinity)

我认为这里所谓的ε^n应该是ε / (2^n)，那样总外测度就是：

ε/2 + ε/4 + ε/8 + ... + ε/(2^n) + ... = lim ε (1-2^(-n)) = ε

这个结果才符合他的下文。

(7) Lebesgue then defined the inner measure of a set E bounded by the interval [a,b] in terms of the outer measure of the set of points contained in this interval minus the set of points contained in the set E (denoted by "[a,b] - E"). This definition can be written as

m_i(E) = (b - a) - m_e([a,b] - E)

where m_i is the inner measure and m_e is the outer measure.

At this point, Lebesgue showed that "the inner measure is never greater than the outer measure" and then stated the key definition of the measurability of a set:

"Sets for which the inner and outer measures are equal are called measurable and their measure is this common value".

这里内外测度殊途同归的"measure"就是著名的Lebesgue测度，Rudin老先生费了半天劲也没让读者抓住要害的“可测性”概念也终于真相大白！我们运用此定义就可推出[0,1]区间内的有理数集合之Lebesgue测度为0，而其内的无理数集合之Lebesgue测度为1。在定义完“可测集合”之后Lebesgue又继续前进，以此为基础定义了“可测函数”：

"We say that a function f, bounded or not, is measurable if, for any c < d, the set {x|c<f(x)<d} is measurable".

It is fair to say that any function ever considered prior to 1900 belonged to the family of Lebesgue-measurable functions. It was really a huge collection. Now he is ready to define the famous Lebesgue Integral.

幸运的是，欧美数学界确实有这样具有高可信度的优秀数学教育家，William Dunham教授无疑是其中的典范之一。我在2005年读了从图书馆借出的"Journey Through Genius"几个章节之后，就果断在Amazon买了Dunham教授全部四本著作--- "Journey Through Genius", "Euler---The Master of Us All", "The Mathematical Universe", 和最新的"The Calculus Gallery"。
＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝

谢谢！
一定找时间拜读：）

(8) Riemann's integral of a bounded function f started with a partition of the domain [a,b] into tiny subintervals, built rectangles upon these subintervals whose heights were determined by the functional values, and finally let the width of the largest subinterval shrink to zero. By contrast, Lebesgue's alternative was predicated upon an idea as simple as it was bold:

Partition not the function's domain, but its range!

In Stephen Hawking's words: "I remember once hearing a story about an historian of Babylonian astronomy whose colleagues asked him for help interpreting an ancient astronomical manuscript. He stroked his beard for a while and pondered the odd symbols on the manuscript. Then suddenly, he began to explain the manuscript to his colleagues as he rotated it 90 degree from portrait mode to landscape mode. Henri Lebesgue made just such a change in perspective for the theory of integration. ... Lebesgue's great insight was to partition the y axis, rather than the x axis, in much the same way that Cauchy and Riemann had partitioned the x axis."

Lebesgue的这一90度"脑筋急转弯"堪称神来之笔！那些不规则函数在y轴上的频繁剧烈振荡曾将Riemann搞得满头大汗，也只有像他这样无所畏惧的数学奇才有能力理出一个大概头绪，他的结论是这些振荡必须是"可控的"。但是如何精确定义"可控"又是一个新的难题，而Lebesgue的神来妙手将这些困难完全绕过，他的积分方法不再分割x轴，而是分割y轴。这样组成积分的那些小矩形的一边长度就是分割y轴后的一系列函数值，另一边则是x轴上可测点集的Lebesgue测度(而在x轴上只有集合的测度，而根本不存在数值振荡问题)，这两个数值相乘就是小矩形的面积。把这些面积求和并在y轴分割的小区间最大宽度(ε)趋于0时求极限，如果这一极限值唯一存在，它就是著名的Lebesgue积分。

Lebesgue本人在1966年出版的一本科普小册子"Measure and the Integral"中做了一个形象生动的类比：

He imagined a shopkeeper who, at the end of a day, wishes to count the total daily revenue. One option is for the merchant to "count coins and bills at random in the order in which they came to hand". Such as merchant, who Lebesgue called "unsystematic", would add the money in the sequence in which it was collected: a dollar, a dime, a quarter, another dollar, another quarter, and so on. This is like taking functional values as they are encountered while moving from left to right across the interval [a,b]. This is the spirit of Riemann's integral.

On the other hand, would it not be preferable for the merchant to ignore the order in which the money arrived and instead group it by denomination(货币面额)? The calculation of the day's revenue would then be simple: multiply the value of the currency (which corresponds to the functional value of the subinterval on y axis) by the number of pieces (which corresponds to the measure of subsets on x axis) and add them up. This is the spirit of Lebesgue's integral.

Rudin一类的数学专家恐怕穷其一生也想不出这样巧妙的类比来启发学生的思路。数学的思想就应该是鲜活的，而不是由无数概念，定理，和证明来堆砌的城堡。而Dunham教授在著述时始终以历史为线索，将各种鲜活的思想有机地联系起来，让人学上几个小时也没有半点枯燥的感觉。最让我击节赞赏的是Dunham教授不时在大段论述推导之后的精辟总结，其语言之优美几乎让人忘记此时是在读一本颇有深度的数学书，在此仅举他在讲完Lebesgue积分后的总结为例：

"With Lebesgue's integral, there was no longer the need to attach restrictive conditions to the derivative, for example, a requirement that it be continuous, in order for the fundamental theorem of calculus to hold. In a sense, then, Lebesgue restored this central result of calculus to a state as "natural" as it was in the era of Newton and Leibniz."

读到这里，我想起了童安格那首著名老歌的结尾句："从梦开始的地方，一切还给自然"，写数学思想时居然能出现如此具有诗意的意境，实在令人心情舒畅！

【尾声】

20世纪初，概率论的各个领域已经取得了大量成果，但是诸如事件和概率等基本概念的定义还存在争议。在19世纪被广泛接受的Laplace关于概率的古典定义在面对Bertrand's paradox等难题时遇到了巨大困难：

http://en.wikipedia.org/wiki/Bertrand's_paradox_(probability)

而Lebesgue积分和Lebesgue测度的应运而生终于让人们意识到：

* 事件的运算与集合的运算完全相似
* 概率与测度具有相同的性质

此时数学界广泛流行的公理化风潮（例如Hilbert将欧氏几何重新严格公理化）促使了前苏联数学家Kolmogorov在1933年提出了著名的公理化结构，使概率论重归严格。而这一公理化结构的最重要的数学基石(mathematical cornerstone)就是Lebesgue测度论，尤其是其中最为关键的"可数可加性"思想(countable additivity or denumerable additivity)！

【总结】

19世纪中叶由德国人Riemann领导的"无数断点函数维新"类似同一历史时期的日本"明治维新"，这次"变法"让某些不连续函数逃离了"连续函数王国"的专制统治和压迫(它们被质疑的"可积性"相当于"合法公民身份")，它们为了自由宁可无家可归，住进"Riemann可积函数"这个带有一定民主色彩的四处飘零的"君主立宪国"。但是在这个酷似"难民营"的国度中，类似Dirichlet函数这样的怪胎依然不被社会所接纳，无法取得"可积函数身份证"这样的基本"人权",。于是这些少数受压迫者意识到，只有彻底推翻君主制，建立一个完全民主的"共和国"，它们才有真正的容身之地。

20世纪初由法国人Lebesgue领导的"可测函数革命"则类似同一历史时期由孙中山发动的"辛亥革命"，这场"变革"成功地建立起一个极具包容性的"可测函数共和国"，终于让Dirichlet函数这样的怪胎也成为合法的可积函数。这样，几乎所有在当时已知的函数都能在这个自由民主平等的家园中找到自己的位置。

"可测函数共和国"的建立具有重要的历史意义，它将"函数可积性"彻底从"函数连续性"中独立出来，也为其后由俄国人Kolmogorov领导的“概率论公理化革命”铺平了道路，其重要性绝不亚于当年美利坚合众国成功摆脱大英帝国的统治。:-)

存下来再细看

Rare masterpiece! A brilliant exposition of an otherwise very dry subject.

Happy new year.

非常精彩。Omni 老兄的求知精神值得我们学习。现在的数学教科书的确忽视了接受数学训练较少的读者。自 Hilbert， Bourbaki 以下，作者都注重逻辑上的完备，所以难免牺牲很多原始的启发性想法。如果两头兼顾，书本的厚度将超过读者的忍受极限，所以只好牺牲生动性而追求严格。

比如 Lebesgue 可测集的定义，原始的想法的确是外测度和内测度相等，然而这只对有界集有意义，一般的无界集合外测度和内测度都是无穷，无法比较大小。所以才有了现在书本上的可测集定义，即，只定义外测度，而 E 可测当且仅当它和它的补集把任何“试验集”分成两部分，使得这两部分的外测度之和是原试验集的外测度。这样本质上是用 “试验集” 把无界的情况归结于局部，从而达到逻辑上的完备。当然这并不是所有刚开始接触实变函数的读者都能体会的。

如果说一个人的数学修养像一棵树，那么 “根深” 和 “叶茂” 就是相辅相成的两个方面。有的时候的确是需要在第一遍学习的时候囫囵吞枣，而通过做习题和学习更多的课程而慢慢体会旧知识的深刻的背景和涵义。也就是说，有时候 “叶茂” 以后才能 “根深”。

Omni老兄说得太好了.勒贝格测度确实是了不起.在函数类中,由有界函数--收敛函数--黎曼可积函数--连续函数--可微函数,一路下来,是不是后一函数都是前一函数的子集呢?有请哪位大虾讲一下.

看完文章,在概率统计方面又长了一大截:)

……连续函数--可微函数,一路下来,是不是后一函数都是前一函数的子集呢?
===============================================================
连续未必可微，可微未必连续。
请认真阅读这个网页：http://www.imb.xjtu.edu.cn/courshow/wljc/chapter5/03/syjn.htm

……连续函数--可微函数,一路下来,是不是后一函数都是前一函数的子集呢?

##########
在复变函数情况下，可微就必定连续，反之不然。
嘻嘻