NB: there -is- a place for epsilon-deltas in math:
that is, epsilon-deltas are -necessary- to understand
-uniform continuity- and -uniform convergence-
A key example illustrating this was Cauchy's `proof' that the limit
of continuous functions is continuous: uniformity is implicit.
(in "Focus", 2004-Jan, a professor argues this)

That is, you really -cannot- understand uniformity or properties that
rely on it w/o epsilons and deltas.
...but you should introduce these -exactly- when you need these
uniformity properties -and not before-!
(note also that I think -oscillation- is in many ways an easier
 concept than deltas and epsilons, b/c it replaces the forall/exists
 by a limit, namely the oscillation at that point)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(wow -- I just thought -exactly- the same thing again today
 as the above; here's an elaboration):
- epsilon-deltas are just -used- as painful -formal manipulations-;
  if you want to teach -formal- manipulations, I think -algebra- is a
  much better place to do it (much less painful); analysis
  manipulations are -hard-
    (esp. if they're meaningless)

  epsilon-delta -can- be semantic, meaningful, but that takes -far-
  more work, and it simply isn't done;
  it's a big stretch to show how they're meaningful,
  and I think "oscillation" does it better (b/c it eliminates a few of
  the symbols)

that is, epsilon-deltas are perfect for "weeding out" anyone who
doesn't enjoy painful manipulations.
[further, there's 2 or 3 stupid tricks, esp. for quadratics;
 these can be made meaningful (in the context of "showing
 multiplication is continuous"), but they generally aren't]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Further, I think -lipschitz- functions are an important clarifying
example (one can introduce these as generalizations of $C^1$ functions
on compact sets): you can always pick $\delta=\epsilon/K$.

Note also that -epsilons- in the definition of -limit- are fine, and
good for rigor and intuition.
Note however that one should discuss -limits- with epsilons
-well before- doing continuity rigorously.

--------------
NB: I get choked up and angry every time I deal with this;
someday I'll have a wonderfully written up form for my students,
and a nice enough form for Anton students, and I'll be at some peace,
but I still cry to think how much pain and torture schoolchildren are
put through -- and it reminds me how much waste there is in education:
10s of billions of hours of beating joy out of kids.
--------------
The epsilon-delta definition of limit (and the epsilon-delta definition of
continuity) is the ``hardest definition in analysis'' (and probably in math),
in the sense that it's so radically different from everything that's come
before, and because you don't work with it or do any real math to justify it.
That is, it is a 200 year jump in abstraction and rigor, from Newton to
Weierstrass, and sticks out like a sore thumb: you don't do anything to
prepare, and it disappears soon after -- it's a traumatic and quickly
forgotten experience.

[There are things that are hard for a good reason, and there are
things that are unnecessarily hard; this is of the latter.]

It's also based on a mistaken view of math, namely that mathematicians come up
with arbitrary axioms and work with them.
Instead, we work with things, and after a while axiomatize them to be able to
study them and their generalizations better -- to understand what
characterizes them and what similar things we may have missed.
(e.g., trying to characterize what we mean by a "manifold" or a
"triangulation" leads to cell complexes, PL-manifolds, topological manifolds,
homology manifolds, poincare duality spaces, etc.)

Axioms reflect our intuition;
\item
  you can't understand axioms without having intutition for the objects
  in question,
\item
  studying (meditating on) the axioms is generally -not- the best way to
  understand the objects and intuition;
  rather, you should look at examples and work with them -- these can help you
  understand and illustrate the axioms, but the axioms alone (without
  examples) won't lead to understanding.

This leads to a theory of pedagogy (following Joe Harris),
that you should work with things before defining them formally,
that you -do- before you -define-.

You needn't be so Joe-ish about it -- you can also make a definition, then
launch immediately into examples and work with them as a way of understanding
and retaining the definitions.

The problem is that it's easier to write definitions and repeat them than to
include clear and illustrative examples.

--------------
If you -have- to do epsilon-delta (e.g., you don't control the
[There are things that are hard for a good reason, and there are
things that are unnecessarily hard; this is of the latter.]

It's also based on a mistaken view of math, namely that mathematicians come up
with arbitrary axioms and work with them.
Instead, we work with things, and after a while axiomatize them to be able to
study them and their generalizations better -- to understand what
characterizes them and what similar things we may have missed.
(e.g., trying to characterize what we mean by a "manifold" or a
"triangulation" leads to cell complexes, PL-manifolds, topological manifolds,
homology manifolds, poincare duality spaces, etc.)

Axioms reflect our intuition;
\item
  you can't understand axioms without having intutition for the objects
  in question,
\item
  studying (meditating on) the axioms is generally -not- the best way to
  understand the objects and intuition;
  rather, you should look at examples and work with them -- these can help you
  understand and illustrate the axioms, but the axioms alone (without
  examples) won't lead to understanding.

This leads to a theory of pedagogy (following Joe Harris),
that you should work with things before defining them formally,
that you -do- before you -define-.

You needn't be so Joe-ish about it -- you can also make a definition, then
launch immediately into examples and work with them as a way of understanding
and retaining the definitions.

The problem is that it's easier to write definitions and repeat them than to
include clear and illustrative examples.

--------------
If you -have- to do epsilon-delta (e.g., you don't control the
syllabus in a calc class, or it's an intro real analysis class), then
-build- to it, in chunks:
- manipulations with one quantifier ($\exists, \forall$)
  by itself; then manipulations with two quantifiers
- scratch work (that's thrown away!), then clean proof
- work with arbitrarily small $\epsilon$
NB: much of the difficulty of $\epsilon$-$\delta$ stuff is b/c it's
done about an arbirary point (same problem with general Taylor series)
Hence it's easier to start by showing that a certain limit is $0$;
this reduces the number of symbols.
Likewise, taking the limit at $\infty$ reduces the number of symbols.
Contrast:
We write $\lim_{x\to a} f(x) = L$
if and only if
$\forall \epsilon > 0$
$\exists \delta > 0$ s.t.
if $0<\abs{x-a}<\delta$
then $\abs{f(x)-L}<\epsilon$

We write $\lim_{x\to +\infty} f(x) = 0$
vs.
$\forall \epsilon > 0$
$\exists N$ s.t.
if $x>N$
then $\abs{f(x)}<\epsilon$

NB: the cost of additional symbols grows superlinearly (in fact, much
worse): learn the easiest -representative- special case, then generalize.

--------------
Note that this is also a problem in many classes, for instance real analysis,
where you spend ages working with axioms and definitions, and don't get a feel
for the objects.
You're also frequently missing the big picture.

I.e., rather than abstractly defining a measure and integration against it,
start by just doing it, namely finding torque by integration, and then
clarifying what you're actually doing.

--------------
For teachers: note that the definition of a function being continuous
is that it's continuous -at each point-, while the youthful intuition
is that continuity is a -global- property.

Continuity at a point is -weird- and feels -unnatural-, but it's
-very- powerful (e.g., think of the Peano curve, or the nowhere
differentiable, everywhere continuous functions).
It's not arbitrary, but it's a serious step.
--------------
That said, how can you understand the epsilon-delta definition of limit?
The key point is that they -quantify- and name the qualitative features.
How close? Within epsilon.

(key in notes from home: basically, gloss this:)
We write $\lim_{x\to a} f(x) = L$
if and only if
$\forall \epsilon > 0$
$\exists \delta > 0$ s.t.
if $0<\abs{x-a}<\delta$
-- and at this stage, it's -hard- to learn stuff w/o lots of
computations!
--------------
FIXME: write out one or two carefully worked out epsilon-delta proofs:
show the scratch work (as it's done, i.e., one step at a time),
-then- show the clean answer.
(ideally you've actually done some of these "scratch work, then clean
proofs" before)
Doing epsilon-delta proofs for specific limits and functions
is very little content, and almost $100\%$ protocol, as Danny Madanes says.

They -all- go:
Given $\epsilon > 0$
take $\delta = \epsilon/5$ (or $\delta=\min(1,\epsilon/17)$, etc.)
If $\abs{x-a}<\delta$
...
...computations...
...
so $\abs{f(x)-L}<\epsilon$

If they do not go -exactly- like that, give -no- credit.
--------------
There is -some- value in epsilon-delta proofs:
you learn to be very precise, pedantic, and methodical,
which is not totally useless: ideally you see -mechanically- where
every step comes in (e.g., in showing that $x^2$ is continuous).
