How to show that Local Neron Height functions are unique

Question

I am reading the proof of Theorem VI.1.1 in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, and I am confused on a step extending a continuous function on $E(K)$ with the $v$-adic topology. Specifically, we have the following set-up:

1. $K$ is a number field, $E(K)$ is the set of $K$-rational points of an elliptic curve $E$ defined over $K$.
2. $v$ is a place of $K$, and the $v$-adic topology is defined on $E(K)$ by taking a base of open sets at a point $(x_0,y_0)$ to be all points of $E(K)$ that are $\epsilon$-closed to $(x_0,y_0)$ in both coordinates under the absolute value $| \cdot |_v$ if $(x_0,y_0)$ is not the origin for $E$, and the set of all points with $|x|_v > 1/\epsilon$ at the origin.

We want to prove that the following properties uniquely define a function $\lambda : E(K) \setminus \{\mathcal{O}\} \to \mathbb{R}$:

1. $\lambda$ is continuous on $E(K) \setminus \{\mathcal{O}\}$ and is bounded on the complement of any $v$-adic neighborhood of $\mathcal{O}$.
2. The limit $\lim_{P \to \mathcal{O}} \{\lambda(P) + \frac{1}{2}v(x(P))\}$ exists.
3. For all $P \in E(K)$ with $[2]P \neq \mathcal{O}$, $$\lambda([2]P) = 4\lambda(P) + v((2y + a_1 x + a_3)P) - \frac{1}{4}v(\Delta),$$

where $a_1, a_3, \Delta$ are all defined as usual by choosing a Weierstrass equation for $E$ over $K$.

To prove uniqueness, Silverman supposes we have two such functions $\lambda$ and $\lambda'$, and then considers $\Lambda = \lambda - \lambda'$. He shows that $\Lambda([2]P) = 4\Lambda(P)$ provided that $P \neq \mathcal{O}$, and then says

But the points satisfying $[2]P = \mathcal{O}$ forma discrete subset of $E(K)$..., so by continuity $\Lambda([2]P) = 4\Lambda(P)$ holds for all $P$.

It seems like he wants to apply a sort of density argument: since $\Lambda([2]x)$ and $\Lambda$ are continuous, $\Lambda([2]x) - 4\Lambda(x)$ is continuous, and since this function takes the value $0$ at all $P$ except the finitely many $P$ with $[2]P = \mathcal{O}$, it must also take the value $0$ at the remaining points.

The issue I'm stuck on is whether we know that the $v$-adic topology on $E(K)$ is connected in order to apply such a result? Couldn't the points with $[2]P = \mathcal{O}$ be isolated points in the $v$-adic topology on $E(K)$, so that the continuity of $\Lambda$ doesn't help us determine its value at those points? Even worse, couldn't we potentially find an $E$ and $K$ such that $E(K) = E[2]$, so that the $2$-torsion points are all of the points that we've defined our topology on?