Monthly Archives: April 2017

Cauchy Regular and continuity

Definition. Let \left(X,d_{X}\right),\left(Y,d_{Y}\right) be metric spaces and f:X\rightarrow Y a function. We say f is Cauchy regular provided that any Cauchy sequence \left\{ a_{n}\right\} in X, \left\{ f\left(a_{n}\right)\right\} is a Cauchy sequence in Y .

Note that if f is uniformly continuous on X, then it is Cauchy regular. Indeed, let \left\{ x_{n}\right\} be a Cauchy sequence in X and let \varepsilon>0 be given. Since f is uniformly continuous, there exists \delta>0 such that d_{X}\left(x,y\right)<\delta implies d_{Y}\left(f\left(x\right),f\left(y\right)\right)<\varepsilon.

Since \left\{ x_{n}\right\} is Cauchy, there exists N such that n,m\ge N implies d_{X}\left(x_{n},x_{m}\right)<\delta. So d_{Y}\left(f\left(x_{n}\right),f\left(x_{m}\right)\right)<\varepsilon. Hence \left\{ f\left(x_{n}\right)\right\} is Cauchy in Y.

One may ask whether a Cauchy regular function is continuous. It is clearly true.

Proposition. Let \left(X,d_{X}\right),\left(Y,d_{Y}\right) be metric spaces
and f:X\rightarrow Y a Cauchy regular function. Then f is continuous
on X.

Proof. Suppose not. There exists x_{0}\in X such that f is not continuous at x_{0}. Then there exists \varepsilon_{0}>0 such that for any \delta>0, there exists x\in X such that d_{X}\left(x,x_{0}\right)<\delta but d_{Y}\left(f\left(x\right),f\left(x_{0}\right)\right)\ge\varepsilon_{0}.

By taking \delta=\frac{1}{n}, we can choose x_{n} in X such that x_{n}\rightarrow x_{0} in X. Now define y_{n}=\begin{cases} x_{n} & \text{odd}\\ x_{0} & \text{even} \end{cases}. Since x_{n} is convergent, \left\{ y_{n}\right\} is Cauchy. So \left\{ f\left(y_{n}\right)\right\} is Cauchy. But

    \[ d_{Y}\left(f\left(y_{2n}\right),f\left(y_{2n+1}\right)\right)=d_{Y}\left(f\left(y_{2n}\right),f\left(x_{0}\right)\right)\ge\varepsilon_{0}, \]

shows \left\{ f\left(y_{n}\right)\right\} is not Cauchy, a contradiction.

So Cauchy regular is continuous. Here we note that we didn’t used any kind of continuity.

Recall the Heine-Cantor theorem. If X is compact and f:X\rightarrow Y is continuous, then f is uniformly continuous. One may ask what condition on X gurantees a continuous function f is Cauchy regular. In fact, if we impose X is complete, this is enough.

Proposition. Let f:X\rightarrow Y be continuous. If X is complete, then f is Cauchy regular. In this case, continuity and Cauchy regular is
same.

Proof. Let \left\{ x_{n}\right\} be a Cauchy sequence in X. By completeness of X, there is x\in X such that x_{n}\rightarrow x in X.
Then triangluar inequality shows

    \[ d_{Y}\left(f\left(x_{n}\right),f\left(x_{m}\right)\right)\le d_{Y}\left(f\left(x_{n}\right),f\left(x\right)\right)+d_{Y}\left(f\left(x\right),f\left(x_{m}\right)\right). \]

Since f is continuous at x, there exists N such that n,m\ge N implies

    \[ d_{Y}\left(f\left(x_{n}\right),f\left(x\right)\right)<\frac{\varepsilon}{2}\quad\text{and}\quad d_{Y}\left(f\left(x\right),f\left(x_{m}\right)\right)<\frac{\varepsilon}{2}. \]

Hence there exists N such that n,m\ge N implies

    \[ d_{Y}\left(f\left(x_{n}\right),f\left(x_{m}\right)\right)<\varepsilon. \]

So f is Cauchy regular.

Example. Define f:\mathbb{Q}\rightarrow\mathbb{R}

    \[ f\left(x\right)=\begin{cases} 0 & \text{if }x^{2}<2\\ 1 & \text{if }x^{2}\ge2. \end{cases} \]

Then f is continuous \mathbb{Q}. But this function is not Cauchy regular since it cannot be extended to \mathbb{R} as a continuous function. The example does not contradict our previous theorem since \mathbb{Q} is not complete.

There is a continuous function f which is Cauchy regular, but not uniformly continuous. Consider f:[0,\infty)\rightarrow\mathbb{R} defined by f\left(x\right)=x^{2}. Then it is Cauchy regular, but not uniformly continuous.

Definition. A metric space \left(X,d\right) is Cauchy precompact if every sequence admits a Cauchy subsequence.

Proposition. If X is Cauhcy-precompact and f:X\rightarrow Y is Cauchy regular, then f is uniformly continuous.

Proof. Suppose not. Then there exists \varepsilon>0 and sequences \left\{ a_{n}\right\}, \left\{ b_{n}\right\} such that d_{X}\left(a_{n},b_{n}\right)\rightarrow0
but d_{Y}\left(f\left(a_{n}\right),f\left(b_{n}\right)\right)\ge\varepsilon. Since X is Cauchy precompact, there exists a Cauchy subsequence \left\{ a_{n_{k}}\right\} of \left\{ a_{n}\right\}. Again by Cauchy-precompactness, there exists a Cauchy subsequence \left\{ b_{m_{k}}\right\} of \left\{ b_{n_{k}}\right\}. Define \left\{ c_{k}\right\}
by a_{m_{1}}, b_{m_{1}},a_{m_{2}},b_{m_{2}},\dots. One may check it is Cauchy sequence. Then if k is odd, then

    \[ d_{Y}\left(f\left(c_{k+1}\right),f\left(c_{k}\right)\right)=\left|f\left(a_{m_{k^{\prime}}}\right)-f\left(b_{m_{k^{\prime}}}\right)\right|\ge\varepsilon_{0}. \]

But \left\{ f\left(c_{k}\right)\right\} is Cauchy, a contradiction. Hence f is uniformly continuous on X.

Teaching at the University Level

다음은 Johns Hopkins 대학의 Steven Zucker교수가 AMS Notices(1996년 8월호)에 쓴 Article 가운데 있는 대학 신입생 Orientation 내용이다. 그는 이를 통하여 비 정상적으로 행하여지고 있는 대학의 Calculus 및 여타 수학 교육의 해결방안을 제시하고 있다. (Steven Zucker, Teaching at the University Level, Notices AMS, pp. 863 – 865 참조)
옮긴이: 김영욱 (고려대학교 수학과 교수)

  1. 여러분은 이제 고등학생이 아니다. 여러분들 가운데 대학생활을 아직 잘 인식하지 못한 대부분 학생들은 고등학교식으로 수업하고 공부하는 것은 잊어버리고 대학교식으로 공부하는 법을 배워야한다. 이는 어려운 일일지는 모르나, 조만간에 해야할 일이기에 빠를수록 좋다. 우리 목표는 단순히 강의시간에 강의한 내용을 여러분이 되풀이 할 수 있게 되는 수준을 넘어선다.
  2. 강의하는 속도는 고등학교때의 두배 내지 세배가 될 것이다. 그 뿐만이 아니라, 배우는 내용을 고등학교때 보다 더 잘 응용할 수 있도록 하려고 한다. 특히 배운 내용을 새로운 문제에 잘 적용하는 능력을 키우려고 한다.
  3. 강의시간은 매우 소중하므로 효과적으로 써야한다. 강의시간 모든 것을 배울 수는 없다. 강의된 내용을 배우는 것은 여러분의 몫(책임)이다. 강의내용의 대부분은 강의실 밖에서 배워야 한다. 여러분은 강의 한시간당 두시간씩 추가로 시간을 쓸 각오가 되어 있어야 한다.
  4. 강사(교수)의 책임은 주로 강의 내용의 골격을 잡아주는 일이며, 가끔은, 여러분이 강의 내용을 이해하고 방법을 배우도록, 특수한 상황을 설명하기도 한다. 여러분이 모든 가능한 상황이나 문제 유형을 익히도록 여러분을 풀그림program하거나 여러분이 이러한 것을 제대로 배워나가는가를 지켜보는 것은 교수가 하는 일이 아니다.
  5. 여러분은 교과서를 읽어서 이해해야 한다. 교과서에는 이 과목에서 가르치는 내용이 상세하게 설명되어 있다. 또 많은 문제들이 풀려있고, 강의시간중에 설명한 내용에다 덛붙여 내용을 이해하는데 도움이 되도록 활용해야 한다. 교과서는 소설책이 아니다. 따라서 교과서를 읽는 것은 때로 천천히 읽기도 해야하고 조심스럽게 내용을 짚어보기도 해야 한다. 그러나 이러한 방법은 여러분이 여러분에게 맞는 속도로 책을 읽을 수 있다는 장점도 가지고 있다. 연필과 종이를 준비하고 교과서 내용을 풀어보며, 설명을 건너뛴 부분도 채워 넣는다.
  6. 언제 교과서를 보느냐에 따라 다음 두가지 경우dichotomy가 생긴다:
    1. (대부분 학생들에게 추천하는 방법) 강의시간에 강의를 듣기 이전에 강의될 내용을 미리 읽어 둔다. 그러면 속도가 빠른 대학의 강의를 따라가기 쉬울 것이다.
    2. 미리 교과서를 읽지 않았으면, 강의시간중에 알아들을수 있는 만큼 최대한 기억한다.(일반적인 idea를 먼저 알아내거나/내고 강의 노트를 최대한 자세히 한다.) 그리고는 강의 후에 교과서로 다시 공부하면서 시간중에 배운 내용이 정리되기를 바란다.

 

원문

 

What follows is what an entering freshman should hear about the academic side of university life. It is distilled from what I’ve learned and written concerning the need for academic orientation as a result of having been the instructor of 110.109 (Calculus II: Physical Sciences) in the fall semester for three consecutive years. The underlying premise, whose truth is very easy to demonstrate, is that most students who are admitted to a university like JHU were being taught in high school well below their level. The intent here is to reduce the time it takes for the student to appreciate this and to help him or her adjust to the demands of working up to level.

  1. You are no longer in high school. The great majority of you, not having done so already, will have to discard high school notions of teaching and learning and replace them by university-level notions. This may be difficult, but it must happen sooner or later, so sooner is better. Our goal is more than just getting you to reproduce what was told to you in the classroom.
  2. Expect to have material covered at two to three times the pace of high school. Above that, we aim for greater command of the material, especially the ability to apply what you have learned to new situations (when relevant).
  3. Lecture time is at a premium, so it must be used efficiently. You cannot be “taught” everything in the classroom. It is your responsibility to learn the material. Most of this learning must take place outside the classroom. You should be willing to put in two hours outside the classroom for each hour of class.
  4. The instructor’s job is primarily to provide a framework, with some of the particulars, to guide you in doing your learning of the concepts and methods that comprise the material of the course. It is not to “program” you with isolated facts and problem types nor to monitor your progress.
  5. You are expected to read the textbook for comprehension. It gives the detailed account of the material of the course. It also contains many examples of problems worked out, and these should be used to supplement those you see in the lecture. The textbook is not a novel, so the reading must often be slow-going and careful. However, there is the clear advantage that you can read it at your own pace. Use pencil and paper to work through the material and to fill in omitted steps.
  6. As for when you engage the textbook, you have the following dichotomy:
    1. [recommended for most students] Read for the first time the appropriate section(s) of the book before the material is presented in lecture. That is, come prepared for class. Then the faster-paced college-style lecture will make more sense.
    2. If you haven’t looked at the book beforehand, try to pick up what you can from the lecture (absorb the general idea and/or take thorough notes) and count on sorting it out later while studying from the book outside of class.