# Sergei Yakovenko's blog: on Math and Teaching

## Monday, January 23, 2012

### Incomplete posting: differentiability of maps and functions

Filed under: Uncategorized — Sergei Yakovenko @ 8:33

Towards the next lecture on Jan 24, 2012, the notes for this and the previous lecture. It will be completed and uploaded once the subject is treated in full, then the link will be replaced.

## Sunday, January 22, 2012

### תרגיל בית מס’ 4

שלום,

מצורף קישור לתרגיל בית רביעי ואחרון.

תאריך הגשה: 31 בינואר. הפעם לא יתקבלו איחורים.

בהצלחה,

דימה

## Tuesday, January 17, 2012

### פתרון של תרגיל בית מס’ 2

בלינק זה תמצאו את הפתרון של תרגיל בית מס’ 2

בברכה,

דימה

## Properties of continuous functions. Basic notions of topology

The standard list of properties of functions, continuous on intervals, includes theorems on intermediate value, on boundedness, on attainability of extremal values etc.

We explain that these results are manifestations of the following phenomenon. There are several properties of subsets of $\mathbb R^1$ (and, in general, arbitrary subsets of the Euclidean space), which can be defined using only the notions of limit and proximity. Such properties are called topological. Examples of such properties are openness/closeness, connectedness (arc-connectedness) and compactness.

The general principle then can be formulated (vaguely) as follows: the topological properties are preserved by continuous maps (or their inverses).

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