In its simplest form, the principle of permanence states that, given any analytic function 
defined on an open (and connected)
set 
of the complex numbers 
, and a convergent sequence 
which along with its limit 
belongs to 
, such that 
for all 
, then 
is uniformly zero on 
.
This is easily proved by showing that the Taylor series of 
about 
must have all its coefficients equal to 0.
The principle of permanence has wide-ranging consequences. For example, if 
and 
are analytic functions
defined on 
,
then any functional equation of the form
 |
that is true for all 
in a closed subset of 
having a limit point in 
(e.g., a nonempty open subset of 
) must be true for all 
in 
.
