A number 
is called a barrier of a number-theoretic
function 
if, for all 
, 
.
Neither the totient function 
nor the divisor function 
has a barrier.
Let 
be an open
set and 
,
then a function 
is called a barrier for 
at a point 
if
1. 
is continuous,
2. 
is subharmonic
on 
,
3. 
,
4. 
(Krantz 1999, pp. 100-101).
