A function f is differentiable if it has a derivative just as
in calculus or, as in high school,
a tangent line. If the derivative, another function, is
differentiable in the same sense,
then the function f is twice
differentiable. For any positive integer n, in the same way, can say what it means
for function f to be n-times
differentiable. Finally, if
for each positive integer n,
function f is n-times
differentiable, then function
f is infinitely differentiable.

Why do we care about function
f being infinitely differentiable? Because that's
necessary for function f
to have a Taylor series
expansion. Why care? It says
we have a start on a ready made way to
approximate f as accurately
as we wish with, also some
good, first-cut information
on the function from if only
from the first few terms.
With more, as Milnor mentioned,
the function f might be
analytic which means that
it is equal to its Taylor
series expansion.

In R^n, a set is open if
it contains a little ball
around each of its points.
So, in R the interval (0,1)
is open. A set C a subset of
R^n is closed if and only
if R^n - C is open. Open
sets have some nice properties;
so do closed sets.

E.g., we know from calculus
from uses of deltas and epsilons
what a continuous function is.
Well we can generalize to topoloty, Milnor's subject:
Quite generally, function f
is continuous if
the inverse image of
each open set in the range
set of f is open in the
domain set of f --
and this definition works with
astounding generality.
Topology is heavily interested
in continuity.

In R^n, a each bounded, closed
set is compact which is
really a generalization of
it being finite and finger
lick'n good: E.g., in calculus,
each f: R --> R continuous
on a compact set has a
Riemann integral on that set.
Biggie stuff.

Can take open and closed off to,
say, Banach space (complete
normed vector space)
and the famous open mapping
and closed graph theorems.

For the result I discussed,
open and closed were
central concepts.

The Kuhn-Tucker conditions
are in optimization. So,
imagine you are in a
dark cave with
walls and a floor that is
not flat. If you put down
a ball and it rolls, then
you are not at the bottom.
So, a necessary condition
for being at the bottom
is that the ball not roll.
For that it may be that
the floor would let
the ball roll but the walls
stop it. So, when do the
walls do this? Roughly the
normal vectors of the walls
in contact with the marble
block the motion for the
marble to roll lower.
But for the simple Kuhn-Tucker
equations, in terms of
gradients, work, the
walls must be sufficiently
nice, that is, satisfy
some constraint qualifications.
There are lots of constraint
qualifications, some easier
to verify than others.
In nearly all routine engineering
problems, the constraints
do satisfy some, typically
several, constraint qualifications.
But if want to dig into
lots of details, look
at pathological cases,
etc. then can wonder about
constraint qualifications.
Kuhn and Tucker have a
statement of some constraint
qualifications (CQ), and
so does Zangwill. Are the
two CQs equivalent? One
direction was no. I showed
that the other direction was
also no and, thus, that the
two CQs are independent,
with neither implying the other.
My proof was by counterexample,
from my result, and was
bizarre and tricky, which
is likely why it was new.
For my counterexample, I
used my result about
infinite differentiability,
but that was much more
than I needed. I showed
infinite differentiability
only out of curiosity --
just wondered if it was true.

The result is curious
because intuitively an
infinitely differentiable
function is supposed to
be very smooth (Milnor
gave a precise definition
but I was just being intuitive)
but some of the level sets
can be wildly irregular,
e.g., the Mandelbrot
set and Brownian motion.
E.g., Brownian motion
never has a tangent line,
not even a first derivative,
but it is the level set
of an infinitely differentiable
function. So, the juxtaposition
is curious.

Yet again
a math guy stands in awe
of reality that math
discovers one little theorem
at a time.

So, now you know more about
what the heck we are talking
about!