This is is a question which will help me ensure I have understood complex analysis and analytic continuation.
We start with an analytic function $f(z)$, holomorphic in the entire complex plane for simplicity.
If we then have a candidate analytic function $g(z)$ which agrees with $f(z)$ at two isolated points $z_1$ and $z_2$.
Question 1: Agreement between $f(z)$ and $g(z)$ at $z_1$ and $z_2$ is insufficient to say they are equivalent, or one is an analytic continuation of the other. Is this correct?
I know (I think) that if there is agreement at every point along a path between $z_1$ and $z_2$, that is sufficient.
Question 2: If there is agreement between $f(z)$ and $g(z)$ at $z_1$ and $z_2$ regarding both their values, and also their complex derivates at those points, then this is sufficient. Is this correct?