Chapter 1: Limits.

Section 1.3 and 1.4 (More Complicated Limits and Continuity):

Some worked homework examples:

Now, try the homework from section 1.4.

*** The formal definition of continuity, and two strange examples mentioned earlier.

(The material that follows is NOT required for the course, but you might find it interesting, and we'll come back to these examples a few times in the course.)

 Here are some notes from a talk showing why we (occasionally) need the very precise definition of limits and continuity we get from the epsilons and deltas:

* The first of these two strange examples is often called Dirichlet's Function, or the indicator (or truth or characteristic) function for the rationals.  The second (more interesting) example, is usually called the "ruler function," but also known as Thomae's Function.  For notions of the sizes of rational and irrational numbers, as well as a definition of "almost everywhere," see, for example, the articles from PlanetMath.org:

http://planetmath.org/?op=getobj&from=objects&name=ProofThatTheRationalsAreCountable

http://planetmath.org/encyclopedia/Countable.html

http://planetmath.org/encyclopedia/CantorsDiagonalArgument.html

http://planetmath.org/encyclopedia/AlmostSurely.html

(By the way, PlanetMath is an excellent resource, if you're just interested in exploring math topics on your own.)

Next time, FINALLY SOME CALCULUS!!!!