Let $K$ be a compact Hausdorff space such that the set $D$ of isolated points in $K$ is countable and dense in $K$. Consider the linear subspace $A$ of $C(K)$ consisting of those functions $f\in C(K)$ such that $f$ is constant on $K\setminus D$ and the set $\{x\in D\colon f(x)= 0\}$ is finite or its complement in D is finite.
Is $A$ a closed subspace of $C(K)$?