## 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.

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 |