A function $f:A \subset \Bbb C \to \Bbb C$ is said to be analytic if it is locally given by a convergent power series i.e. every $z_o \in A$ has a neighbourhood contained in $A$ such that there exists a power series centered at $z_o$ which converges to $f(z)$ for all $z$ in that neighbourhood.
Does that mean that this power series should also converge to $f(z_o)$ at $z_o$?
$z_0$ is in the neighborhood centered at $z_0$ (here I mean that the neighborhood does contain $z_0$), so the power series does converge to $f(z_0)$ at $z_0$ by definition. $f$ is hence differentiable at $z_0$ as any power series is differentiable.