What does 'Let $F: X \to Y$ be a holomorphic map defined at $p \in X$, which is not constant' mean?

Question

I refer to Chapter II.4 of Rick Miranda - Algebraic curves and Riemann surfaces.

There's this statement

Let $F: X \to Y$ be a holomorphic map defined at $p \in X$, which is not constant.

Here, $X$ and $Y$ are (connected but not necessarily compact) Riemann surfaces.

I find this kind of weird because I don't think a map can be holormorphic on a set without being defined at every point of the set. This is unlike the case for 'meromorphic'. Which of the following does this mean?

1. Let $F: X \to Y$ be a non-constant holomorphic map (on all of $X$). Let $p \in X$.

2. Let $F: X \to Y$ be a non-constant map. Let $p \in X$. Suppose $F$ is holomorphic at $p$ (but not necessarily the whole of $X$).

3. Other

Answer 1

When in doubt, check a second source. For example, Donaldson (page 38, Proposition 3) states it as 1.