מצורף קישור לתרגיל בית רביעי ואחרון.

תאריך הגשה: 31 בינואר. הפעם לא יתקבלו איחורים.

בהצלחה,

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בברכה,

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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 (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).

The lecture notes are available here.

שימו לב כי יש לכם את כל הכלים לפתור את התרגיל, למרות שהוא עלול להיראות קשה ממבט ראשון.

תאריך אחרון להגשה: 10 בינואר.

בהצלחה!

In the first lecture we introduce the notion of continuity of a function at a given point in its domain and a very close notion of a limit at a point outside of the “natural” domain.

This notion is closely related to the notion of sequential limit as introduced earlier. This paves a way to generalize immediately all arithmetic and order results from numeric sequences to functions.

The novel features involve one-sided limits, limits “at infinity” and continuity of composition of functions.

The (unfinished) notes, to be eventually replaced by a more polished text, are available here: follow the updates, this temporary link will eventually be erased.

We spend some time considering different flavors of “limit behavior”: stabilization, approximate stabilization etc. A sequence is called converging, if it -stabilizes for any positive accuracy .

To show that passing to a limit “respects” arithmetic operations, we need to work out a bit of “interval arithmetic” with a special attention to the division which may cause problems.

We discuss the weaker notion of a partial limit (accumulation point) and study under what assumptions a unique partial limit is the genuine limit.

Finally, we show that monotone bounded sequences always converge. This is one of the most powerful tools to show that the limit exists when it is not possible to compute it explicitly.

The lecture notes (in pdf) are available here.

מצורף קישור לקובץ בו תמצאו סיכום של הנושאים שלמדנו בשבועות האחרונים.  כל הערה/תיקון יתקבלו בברכה רבה.

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להגשה עד 22 בדצמבר.

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