A bump function is a smooth function with compact support.
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 
, and
3. 
has compact
support.
A smooth function 
satisfying (1)-(3) is a bump function adapted to 
and 
. If 
then by rescaling 
, namely 
, one gets a sequence
of smooth functions which converges
in the sense of distributions to the delta
function, providing that 
is a neighborhood of 0.
