Given any open set 
in 
with compact closure 
, there exist smooth functions
which are identically one on 
and vanish arbitrarily close to 
. One way to express this more precisely is that for any open set 
containing 
, there is a smooth function 
such that
1. 
for all 
and
2. 
for all 
.
A function 
that satisfies (1) and (2) is called a bump function. If 
then by rescaling 
, namely 
, one gets a sequence of smooth functions which
converges to the delta function, providing that
is a neighborhood of 0.
