The following two scenarios I hope will help me fill in gaps in my understanding of analytic continuation.
Scenario 1
We have a function $f(z)$ of a complex variable $z$. It is known to be zero at $z=a$, where $a$ is real for simplicity, but no loss of generality.
$$f(a)=0$$
We also know $f(z)$ is analytic in the domain $D$, and $a\in D$.
Question 1: There is not enough information to say $f(z)=0$ anywhere else in $D$. Is this correct?
Scenario 2:
We have a function $f(z)$ of a complex variable $z$. It is known to be zero for all values of $z$ in the range $[a,b]$, where $a,b$ are real, with no loss of generality.
$$f(z)=0 \text{ for } z \in [a,b]$$
We also know $f(z)$ is analytic in the domain $D$, and all values $[a,b]\in D$.
Question 2: This is enough information to say $f(z)=0$ everywhere in the region of analyticity $D$.
The reason, I suggest, there is enough information is that the gradient of $f(z)$ is zero along the line from $a$ to $b$ and therefore continues beyond it.
Additional Questions:
Question 3: If $f(z)$ is analytic in a subset $D$ of the complex plane $\mathbb{C}$, then it is analytic everywhere in $\mathbb{C}$, except at poles. Is this correct?
Question 4: By analytic, I mean the function is complex differentiable. That is, the cauchy-riemann conditions apply (wiki link). Is this what analyticity is?