Sergei Yakovenko's blog: on Math and Teaching

Wednesday, December 28, 2011

Lecture 10, December 27, 2011

Continuity and limits of functions of real variable

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.


1 Comment »

  1. When dealing with dcpos , one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function f, the image f(D) of a directed set D (i.e. the set of the images of each element of D) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that f preserves directed suprema. Also note that, by considering directed sets of two elements, such a function also has to be monotonic. These properties give rise to the notion of a Scott-continuous function. Since this often is not ambiguous one also may speak of continuous functions.

    Comment by Benito V. Horton — Saturday, February 9, 2013 @ 6:24 | Reply

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