Analytic function and Laurent series

Question

I want to ask a question: Let $B$ is an open subset of $A$ and $f(x)=\sum c_nx^n$ for every $x$ in $B$. Can I conclude that $f(x)=\sum c_nx^n$ for every $x$ in $A$?

Answer 1

No. Take disjoint two disjoint open sets $C$ and $D$. Let $B=C$ and $A= C\cup D$. Can yo see why your assertion fails in this case?

Hint: take $f=1$ on $C$ and $f=0$ on $D$.

If $A$ is connected then the conclusion holds. This is because $g(x) =\sum c_n z^{n}$ defines an anaytic function on $A$. Since $f=g$ in $B$ and $B$ has a limit point in $A$ it follows that $f=g$ on $A$.