Stability of weighted point evaluation functionals

Given $\epsilon >0$, a continuous linear functional $\varphi$ on $C(X)$ is said to be $\epsilon$-disjointness preserving if $\left\vert\varphi(f)\varphi(g)\right\vert\le\epsilon$ whenever $f,g\in C(X)$ satisfy $\left\Vert f\right\Vert _{\infty} =\left\Vert g\right\Vert _{\infty} =1$ and $fg\equiv 0$. In this paper we provide the exact maximal distance from $\epsilon$-disjointness preserving linear functionals to the set of weighted point evaluation functionals.