Summary: Partitions of Unity.
Theorem. Suppose a Rn
and 0 < r < s < . Then there is a smooth function
: Rn
[0, 1]
such that
Ba(r) int{x Rn
: (x) = 1} and spt Ua(s).
Proof. Let f : R R be such that
f(x) =
e-1/x
if x > 0,
0 otherwise.
We have already shown that f is smooth. Let g : R R be such that g(x) = f(x - 2)f(3 - x) for x R;
note that g is smooth, that g vanishes outside (2, 3) and that g is positive on (2, 3). Let h : R R be such
that
h(x) =