Let $D\subseteq \mathbb{C}$ be a domain. Given $z,w\in D$ we define the Harnack distance $\tau_D(z,w)$ as the smallest number such that $\tau_D(z,w)^{-1}h(w)\leq h(z)\leq \tau_D(z,w)h(w)$ for all positive harmonic functions $h$ on $D$.
Question: Why is the Harnick distance continuous? Is it trivial?