family of balls $B_j$ with the property that $\{(3+\delta)B_j\}$ covers $G$, instead of $\{5B_j\}$. In the definition of the set $G$ in Proposition \[prop:coarea\_u\], we may then use $B(x,2(3+\delta)r)$ in place of $B(x,10r)$. We also replace the constant 200 with $4(3+\delta)^2 + \delta$. Following the remainder of the proof and letting $\delta \rightarrow 0$ yields the final value of $\kappa = 216^2\cdot (4/\pi)^2$.

Continuity of $u$ {#sec:continuity_u}
=================

In this section, we strengthen Corollary \[cor:continuity\] by showing that the harmonic function $u$ is continuous on the entire set $Q$. In Theorem 5.1 of [@Raj:16], the continuity of $u$ is proved employing reciprocality condition . In contrast, we do not assume any of the reciprocality conditions in this section.

First, we need a technical fact. This is proved using Proposition 3.1 in [@Raj:16] (which is a re-statement of Proposition 15.1 in [@Sem:96c]) and an induction and limiting argument.

\[prop:curve\_parametrization\] Let $X$ be a metric space and $E \subset X$ a continuum with $\mathcal{H}^1(E) < \infty$. For all $x, y \in E$, there is a 1-Lipschitz curve $\gamma: [0, 2\mathcal{H}^1(E)] \rightarrow E$ such that $|\gamma| = E$, $\gamma(0) = x$, $\gamma(2\mathcal{H}^1(E)) = y$, and $\gamma^{-1}(z)$ contains at most two points for $\mathcal{H}^1$-almost 