Vanishing at infinity is the property of a continuous function 
on a locally compact Hausdorff
space that the set
 |
is compact for every 
(Rudin 1987, p. 70). Such functions
form the Banach space 
with supremum norm
 |
Every continuous function with compact support vanishes at infinity, but the converse need not hold. For example, 
on 
vanishes at infinity and has support all of 
. The continuous functions
with compact support form a dense subspace 
of 
in the supremum norm
(Rudin 1987, p. 129).
