Let $f(z)$ be an analytic function in an open set $U\subset\Bbb{C}$. Recall that an *analytic continuation* of $f$ is a pair $(F,V)$ such that $U\subset V\subset\Bbb{C}$, $F$ is analytic on $V$, and $F(z)=f(z)$ for all $z\in U$. 

My question is, how *stable* is this process? If $\|f-g\|$ is small, are we guaranteed $\|F-G\|$ small in any reasonable sense? If not, are there easy counterexamples? If the answer depends on the choice of norm, I would find that interesting as well.

References gladly accepted in lieu of obvious arguments. Thanks!