לפוסט זה מצורף אוסף בעיות מספר 3. אם עולות שאלות בקשר לבעיות לפני מפגש התרגול הבא, אפשר לשאול כאן.

## Monday, November 30, 2009

### Lecture 5 (Dec 1)

## Continuity

A function defined on a subset is continuous (in full, continuous on ), if it is continuous at each point . Continuity is equivalent to the requirement that the preimage of any open set is open in (i.e., is an intersection between and an open subspace ).

- Continuity at a point is a local property. Automatic continuity at isolated points of .
- Examples of discontinuity points.
- Continuity and arithmetic operations. Continuity of the composition. Continuity of elementary functions.
- Dirichlet function: the ugly beast. Further pathologies.
- Continuity and global properties:
- Continuity and boundedness
- Continuity and existence of roots
- Continuity and monotonicity

- Continuity and functional equations , , , .

## Thursday, November 19, 2009

### אוסף בעיות מספר 2

לפוסט זה מצורף אוסף בעיות מספר 2. אם עולות שאלות בקשר לבעיות לפני מפגש התרגול הבא, אפשר לשאול כאן.

## Wednesday, November 18, 2009

### A sample theorem, a sample proof

Let be a function and a point.

**Theorem.**

(1)

if and only if

(2)

**Proof.**

1. direction:

by (1).

by definition of the limit of sequence .

Therefore .

2. direction: proof by contradiction.

Assume that the claim is wrong.

Then such that the claim is wrong.

Then the claim is wrong.

Then **(*)**.

Let be a sequence of positive numbers and a sequence of points constructed as follows:

- ;
- is obtained from
**(*)**for ; - for all .

Then converges to , and .

Therefore the claim is wrong in contradiction with (2).

___________________________________

**פתרון מילולי**

**שאלות הבנה**:

- בחלק הראשון, מדוע כמעט כל אברי הסדרה x_n נמצאים ב-N?
- בחלק השני, מדוע ניתן לבחור סדרה אינסופית x_n המתכנסת ל-a כך שהתמונות (f(x_n לא נמצאות ב-I?
- מדוע הסדרה המתקבלת (f(x_n אינה מתכנסת ל-A?

## Monday, November 16, 2009

### Lectures 3-4 (Tue, Nov 17, 24; 9:00-11:00)

## Limits of functions and topology of the real line

- Infinity as the
*value*of the limit: . - Functions of real variable: the domain, range etc (recall). Examples: polynomial and rational functions, . Compositions: .
- Limit of a function: one-sided, two-sided. Continuity points.
- Sequential limit
*vs.*“standard” limit: equivalence theorem. - Open and closed subsets on the real line. סביבות וסגור
- Images and preimages. Some algebra of sets: , the same with . Warning: . Operations on infinitely many sets (unions, intersections).
- Local and non-local properties of functions.
- Continuity via open/closed sets.
- Compactness and its implications.

## Saturday, November 14, 2009

### אוסף בעיות מספר 1

לכניסה זו מצורף אוסף הבעיות הראשון. לחלק מהבעיות המאתגרות יותר מצורפות הדרכות – כמובן, יש יותר מדרך אחת לפתור כל בעיה, וכיוונים מקוריים יתקבלו בברכה. אנא קראו את הבעיות בעיון ונסו לפתור לפחות את חלקן.

## Tuesday, November 10, 2009

### Lecture 2 (Thu, Nov 12, 13:30 – 16:30)

## Existence of limits and completeness of the real numbers system

- Monotonicity and its implications.
- Nested intervals and their common point
- Boundedness as another property stable by finite alterations
- Converging subsequense of a bounded sequence
- But why we are so sure that there are no gaps on the real line? And what is a real line?

Construction of the number system: from natural numbers toward scary numbers

- Completion by algebraic operations: from to via . Everything you need to solve linear equations
- Problems with quadratic equations: irrationalities and negative discriminants. An idea of algebraic number.
- Problems with transition to limit: the ubiquitous and much, much more
- Infinite decimal fractions: completion by “adding limits of monotone sequences”.
- Operations with real numbers: ordered field.
**Completeness**“axiom”.

## Wednesday, November 4, 2009

### Lecture 1: Nov 5, 2009

## Limit: the first encounter with infinity

- Introduction and logistics
- Goals of the course
- Infinity: the name of the game in Analysis
- Some examples and counterexamples
- Sets and their subsets that have “the same number” of elements
- Summation of infinitely many terms: success and failure
- Limits in geometry: how to measure the lengths?
- Pathological curves on the plane

- Numerical sequences and their limits: the first real encounter with infinity.
- “
*Almost all*” vs. “*infinitely many*“ - Intervals and operations on them
- Geometric definition of the sequence limit
- “Instrumental” definition of the limit
- “Standard” definition:
- First theorems about limits (limits and arithmetic operations, limit and rearrangements, limit and boundedness, limits and monotonicity)

- “
- Step back: number systems. Integer, rational and real numbers. Sealing the gaps. Completeness

### Caesaria (Rothschild) Programme: Analysis for High School Teachers

This post marks beginning of the new “blogged” (blog-accompanied) course “Analysis for High School Teachers” within the framework of the Caesarea-Rothschild Program. I will place here brief content of the forthcoming lectures and some relevant material. It may come in a variety of electronic formats, of which most popular are pdf and dejavu. e-Readers for these formats are freely available from the internet.

**Blogging rules:**

For those not familiar with the blogging subculture (are there still such people?). You are welcome to leave your comments/questions/remarks next to the relevant posts. Please introduce yourself when commenting. My wet dream is having mathematical discussions between the students attending the course on these pages. Don’t be afraid to express yourself and teach others. I promise not to abuse my rights as a moderator here. This venue for interaction is especially important since it is convenient for students which stay away from the teachers for most of the time.

Any language is accepted, though I **strongly** urge to write in the l.c.d. (= Simple English).

This platform (WordPress) allows for easy insertion of LaTeX code, which makes mathematical discussions here especially pleasant and easy to maintain.

The course will be accompanied by guided seminars led by Gal Binyamini: these seminars will be devoted to discussion of problems and their solutions as well as complimentary material to the main lectures. Gal will also post to this blog.

One of the sources for the course will be the excellent book which equally fascinates both professional mathematicians and high school children. It is in English and you can download an (illegally scanned) copy* strictly for your personal use* 🙂 here (21 Mb in pdf format: beware!). I will also try to upload separate relevant sections of the book next to posts on specific lectures.