No.  The domain of f is all of
R^n so that f(x) has to
be defined for all x in R^n.<p>Then there is the requirement
that f be strictly positive otherwise, that is, in the open set outside C.<p>The result is a bit amazing:
It says that any closed set
can be the level set for 
an infinitely differentiable
function.<p>Can add some interest when
consider how <i>bizarre</i> some
closed set are:<p>So, in the plane
can take C as a sample path of
Brownian motion.  So, there&#x27;s 
an infinitely differentiable
function zero on that sample path
and positive otherwise.<p>Next, the Mandelbrot set is
closed.  So, there&#x27;s an
infinitely differentiable
function positive everywhere
but zero on the Mandelbrot set.<p>Next consider the Cantor set
or Cantor sets of positive
measure.  Same story.<p>So, a really smooth, 
infinitely differentiable, function can have for 
a level set
anything at all considering
that any level set of a 
continuous function is closed.<p>There&#x27;s a famous paper in
mathematical economics by
Arrow, Hurwicz, and Uzawa 
with a question with no
answer, and my work answers
the question.  Yes, Arrow
and Hurwicz won prizes in
economics -- IIRC so far
poor Uzawa has not won
such a prize!<p>For something intuitive, 
consider an infinitely 
differentiable mountain
range, and pour in some water
to make a lake or many
lakes (assume a porous 
mountain) all with the
same altitude.  Then, the lake
boundary is a level set of
an infinitely differentiable
function.  As we know from
Mandelbrot, commonly, roughly
lake boundaries are fractals.<p>So, we&#x27;re fine; everything
holds together:  My result
shows that for Mandelbrot&#x27;s
fractal lakes, the ground
can be smooth, infinitely
differentiable.  Did
Mandelbrot know that?<p>For rational p&#x2F;q for positive
whole numbers p and q with p&#x2F;q
in lowest terms, consider in
R^2 a ray from the origin of
length 1&#x2F;q at angle p radians.
Do this for all such countably
infinitely many rationals.  Then 
the set of all these rays is
closed but is a <i>spiny urchin,</i>
really bizarre.  That&#x27;s the
closed set I used for my
counterexample.<p>Can also apply this to 
zero day detection of
anomalies in server farms
and networks:  So, suppose
collect 20 times a second
numerical data on 10 variables.
Then that vector of 10
variables has a probability
distribution.  Suppose it also
has a continuous density.
Then pour in water to make
lakes -- all the same altitude.
Get a point in a lake, then
declare an anomaly.  The volume 
of land under the lakes is
essentially the false alarm rate.
Pour in more water and get
a higher false alarm rate but
also a higher detection rate.
Here the set where we declare
an anomaly has the highest possible area for the given
false alarm rate, and there&#x27;s
another sense in which the
detector has optimal combinations
for false alarm rate and
detection rate.<p>How &#x27;bout that!