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 .