The big topological results that one uses (and may even prove) in
calculus class are:
``continuous maps preserve connectedness and compactness''.

This is broken up into 3 theorems (at least by Spivak):
- IVT: Intermediate Value Theorem
  This is often mixed up with the MVT: Mean Value Theorem.
  However, the Mean Value theorem is more properly interpreted thus:
  - it's Taylor's theorem (estimate) for degree 1
    ...and this (Taylor's theorem) is rather more subtle than the IVT.
  - it says that an everywhere differentiable function, while it may
    not be continuous, never-the-less satisfies the Intermediate Value
    Property
    ...and this is quite subtle: the derivatives of everywhere
    differentiable functions can be quite weird.

- Boundedness principle
  Every continuous function on a compact interval $f\from [a,b] \to \bR$
  is bounded above (and below)

- Maximum principle
  Every continuous function on a compact interval $f\from [a,b] \to \bR$
  achieves its sup and inf


There's a very easy proof of the IVT using Dedekind cuts, and this is
one of their main attractions.

The boundedness and maximum principle are trivial using your favorite
notion of compactness.
Note that you need to understand sups and infs in some sense to even
state the maximum principle.

However, proofs of all of these are a bit abstract (compared with
lame-o calc class), and it's hard to make exercises about these, so
these are often a waste of time.

Also, this is a great time to get bogged down in the mechanics of
working with sups and infs.
However, this is not such a bad thing to play with. Students -should-
work with these until can work with them -- it's a good part of their
symbolic understanding, and you learn to be much more careful.
(here's a favorite exercise: $\abs{f+g}_\infty \leq \abs{f}_infty +
\abs{g}_\infty$: prove this, and give an example to show equality
needn't hold. It's easy, but important.
More fun: this still holds if your domain isn't compact (IE, even if
your sup/inf isn't achieved). It's no harder to prove, but makes you
think a bit more.)


Here's a few exercises.
These should all be done by students, either as homework, on the exam,
or at the board.

Every odd degree real polynomial has a real root.

The IVT is false for the rationals (give a counterexample. My favorite
is $x^2 - 2$)
