Differential Topology – Lecture 1 by John W. Milnor (1965) [video]
Gee, I can like that!<p>For the set of real numbers
R, he starts with a function<p>f: R --&gt; R<p>such that f(x) is zero for
x &lt;= 0, strictly positive
otherwise, and infinitely differentiable.  There is
such a function, the same or
similar, in an exercise in
Rudin, <i>Principles of Mathematical
Analysis</i>.<p>At one time I used that function
in Rudin
to show that for positive 
integer n and closed subset
C of R^n there exists function<p>f: R^n --&gt; R<p>so that f(x) = 0 for all x in
C, strictly positive otherwise,
and infinitely differentiable.
That result is comparable with
the classic Whitney extension
theorem -- Whitney assumed
a little more and got a little
more.<p>I discovered this result 
for and used it in some work in the constraint
qualifications of the Kuhn-Tucker
conditions:  I constructed
a counterexample that showed
that the Zangwill and Kuhn-Tucker
constraint qualifications
are independent.<p>So, back to Milnor&#x27;s lecture!<p>I want to see if he drags out
the inverse and implicit
function theorems (essentially
the local nonlinear version
of what is standard from
Gauss elimination in
systems of linear equations!).