数学专业英语(Doc版).Word4

数学专业英语－Continuous Functions of One Real Variable

This lesson deals with the concept of continuity, one of the most important and also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concept in an informal and intuitive way to give the reader a feeling for its meaning.

Roughly speaking the situation is this: Suppose a function f has the value f ( p ) at a certain point p. Then f is said to be continuous at p if at every nearby point x the function value f ( x ) is close to f ( p ). Another way of putting it is as follows: If we let x move toward p, we want the corresponding function value f ( x ) to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.

Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x . At each integer we ha

ve what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x approaches 2 from the left, f ( x ) approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x ) does approach f ( 2 ) if we let x approach 2 from the right, but this by itself is not enough to establish continuity at 2. In case like this, the function is called continuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.

In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating look into the exact meaning of continuity. It was not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19th century to examine more carefully the exact meaning of the word “continuity”.

A satisfactory mathematical definition of continuity, expressed entirely in terms of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (17-1857). His definition, which is still used today, is most easily explained in terms of the limit concept to which we turn now.

The definition of the limit of a function.

Let f be a function defined in some open interval containing a point p, although we do not insist that f be defined at the point p itself. Let A be a real number.

The equation

f ( x ) = A is read “The limit of f ( x ) , as x approached p, is equal to A”, or “f

( x ) approached A as x approached p.” It is also written without the limit s

ymbol, as follows:

f ( x )→ A as x → p This symbolism is intended to convey the idea that we can make f ( x ) as close to A as we please, provided we choose x sufficiently close to p.

Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.

Definition of neighborhood of a point.

Any open interval containing a point p as its midpoint is called a neighborhood of p.

NOTATION. We denote neighborhoods by N ( p ), N1 ( p ), N2 ( p ) etc. Since a neighborhood N ( p ) is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p ) by N ( p; r ) if we wish to specify its radius. The inequalities p-r < x < p+r are e

quivalent to –rx whose distance from p is less than r.

In the next definition, we assume that A is a real number and that f is a function defined on some neighborhood of a point p (except possibly at p ) . The function may also be defined at p but this is irrelevant in the definition.

Definition of limit of a function.

The symbolism

f ( x ) = A or [ f ( x ) → A as x→ p ] means that for every neighborhood N1 ( A ) there is some neighborhood N2 ( p) such that

f ( x ) ∈ N1 ( A ) whenever x ∈ N2 ( p ) and x ≠ p (*) The first thing to note about this definition is that it involves two neighborhoods, N1 ( A) and

N2 ( p) . The neighborhood N1 ( A) is specified first; it tells us how clos

e we wish f ( x ) to be to the limit A. The second neighborhood, N2 ( p ), tells us how close x should be to p so that f ( x ) will be within the first neighborhood N1 ( A). The essential part of the definition is that, for every N1 ( A), no matter how small, there is some neighborhood N2 (p) to satisfy (*).

In general, the neighborhood N2 ( p) will depend on the choice of N1 ( A).

A neighborhood N2 ( p ) that works for one particular N1 ( A) will also work, of course, for every larger N1 ( A), but it may not be suitable for any smaller N1 ( A).

The definition of limit can also be formulated in terms of the radii of the neighborhoods

N1 ( A) and N2 ( p ). It is customary to denote the radius of N1 ( A) byε

and the radius of N2 ( p) by δ.The statement f ( x ) ∈ N1 ( A ) is equivalent to the inequality ∣f ( x ) – A∣<ε,and the statement x ∈ N1 ( A) ,x ≠ p ,is equivalent to the inequalities 0 <∣ x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:

The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such that

∣f ( x ) – A∣<ε whenever 0 <∣x – p∣<δ

“One-sided” limits may be defined in a similar way. For example, if f ( x ) →A as x→ p through values greater than p, we say that A is right-hand l

imit of f at p, and we indicate this by writing

f ( x ) = A In neighborhood terminology this means that for every neighborhood N1

( A) ,there is some neighborhood N2( p) such that

f ( x ) ∈ N1 ( A) whenever x ∈ N1 ( A) and x > p Left-hand limits, denoted by writing x→ p-, are similarly defined by restricting x to values less than p.

If f has a limit A at p, then it also has a right-hand limit and a left-hand limit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.

The definition of continuity of a function.

In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not be equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.

Definition of continuity of a function at a point.

A function f is said to be continuous at a point p if

( a ) f is defined at p, and ( b ) f ( x ) = f ( p ) This definition can also be formulated in term of neighborhoods. A funct

ion f is continuous at p if for every neighborhood N1 ( f(p)) there is a neighborhood N2 (p) such that

f ( x ) ∈ N1 ( f (p)) whenever x ∈ N2 ( p). In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:

Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that

∣f ( x ) – f ( p )∣< ε whenever ∣x – p∣< δ

In the rest of this lesson we shall list certain special properties of continuous functions that are used quite frequently. Most of these properties appear obvious when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these statements are no more self-evident than the definition of continuity itself, and therefore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axiom for the real number system.

THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a closed interval [a, b] and assume that f ( a ) an f ( b ) have opposite signs. Then there is at least one c in the open interval (a ,b) such that f ( c ) = 0.

THEOREM 2. Sign-preserving property of continuous functions. Let f be continuious at c and suppose that f ( c ) ≠ 0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ). THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x1 < x2 in [a, b] such that f ( x1 ) ≠ f ( x2 ) . Then f takes every value between f ( x1 ) and f (x2 ) somewhere in the interval ( x1, x2 ).

THEOREM 4. Boundedness theorem for continuous functions. Let f be continuous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤ M for all x in [a, b].

THEOREM 5. (extreme value theorem) Assume f is continuous on a closed interval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = sup f and f ( d ) = inf f .

Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum.

Vocabulary

continuity 连续性 assume 假定,取

continuous 连续的 specify 指定, 详细说明

continuous function 连续函数 statement 陈述,语句

intuitive 直观的 right-hand limit 右极限

corresponding 对应的 left-hand limit 左极限

correspondence 对应 restrict 于

graph 图形 assertion 断定

approach 趋近,探索,入门 consequently 因而,所以

tend to 趋向 prove 证明

regardless 不管,不顾 proof 证明

discontinuous 不连续的 bound 限界

jump discontinuity 限跳跃不连续 least upper bound 上确界

mathematician 科学家 greatest lower bound 下确界

formulate 用公式表示,阐述 boundedness 有界性

limit 极限 maximum 最大值

Interval 区间 minimum 最小值

open interval 开区间 extreme value 极值

equation 方程 extremum 极值

neighborhood 邻域 increasing function 增函数

midpoint 中点 decreasing function 减函数

symmetric 对称的 strict 严格的

radius 半径(单数) uniformly continuous 一致连续

radii 半径(复数) monotonic 单调的

inequality 不等式 monotonic function 单调函数

equivalent 等价的

Notes

1. It wad not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems.

意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.

这里It was not until …that译为“直到……才”

2. The symbol f ( x ) = A means that for every ε> 0 ,there is a δ> 0, such that

|f( x ) - A|<ε whenever 0 <| x – p |<δ

注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.

有时句中there is可换为there exists; such that可换为satisfying; whenever换成if或for.

3. Let…and assume (suppose)…Then…

这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课 Let假设条件的大前提,assume (suppose)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.

4. Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:

f ( x ) approaches A as x approaches p. Approach有时可代以tend to. 如f ( x ) tends to A as x tends to p.值得留意的是approach后不加to而tend之后应

加to.

5. as close to A as we please = arbitrarily close to A..

Exercise

I. Fill in each blank with a suitable word to be chosen from the words given below:

independent domain correspondence

associates variable range

(a) Let y = f ( x ) be a function defined on [a, b]. Then

(i) x is called the ____________variable.

(ii) y is called the dependent ___________.

(iii) The interval [a, b] is called the ___________ of the function.

(b) In set terminology, the definition of a function may be given as follows:

Given two sets X and Y, a function f : X → Y is a __________which ___________with each element of X one and only one element of Y.

II. a) Which function, the exponential function or the logarithmic function, has the property that it satisfies the functional equation

f ( xy ) = f ( x ) + f ( v ) b) Give the functional equation which will be satisfied by the function which you do not choose in (a).

III. Let f be a real-valued function defined on a set S of real numbers. Then we have the following two definitions:

i) f is said to be increasing on the set S if f ( x ) < f ( y ) for every pair of points x and y with x < y.

ii) f is said to have and absolute maximum on the set S if there is a point c in S such that f ( x ) < f ( c ) for all x∈ S.

Now define

a) a strictly increasing function;

b) a monotonic function;

c) the relative (or local ) minimum of f .

IV. Translate theorems 1-3 into Chinese.

V. Translate the following definition into English:

定义:设E 是定义在实数集 E 上的函数,那么, 当且仅当对应于每一ε>0(ε不依赖于E上的点)存在一个正数δ使得当 p 和 q 属于E且|p –q| <δ时有|f ( p ) – f ( q )|<ε,则称f在E上一致连续.