of $s \in (r,2r)$ it holds that $\operatorname*{osc}_{U(x,r)} u = \operatorname{diam}u(U(x,r)) \leq \sum_j \operatorname{diam}u(U_j)$.

Without assuming the reciprocality conditions, it is not clear that the function $u$ is continuous. Nevertheless, Lemma \[lemm:oscillation\] implies a certain amount of continuity for $u$, as we show in the following corollary.

\[cor:continuity\] The function $u$ is continuous at $\mathcal{H}^2$-almost every $x \in Q$.

Inequality implies that $$\limsup_{r \rightarrow 0}  \operatorname*{osc}_{U(x,r)} u \leq \limsup_{r \rightarrow 0} \frac{4r}{\pi}\cdot \frac{1}{r^2} \int_{B(x,2r)} \rho\, d\mathcal{H}^2$$ for all $x \in Q \setminus \partial Q$. Here, $U(x,r)$ is as in Lemma \[lemm:oscillation\]. From basic properties of pointwise densities of measures (see [@Fed:69 Sec. 2.10.19(3)]), the integrability of $\rho$ and local finiteness of $\mathcal{H}^2$ imply that $$\limsup_{r \rightarrow 0} \frac{1}{r^2} \int_{B(x,2r)} \rho\, d\mathcal{H}^2 < \infty$$ for $\mathcal{H}^2$-almost every $x \in Q$. The result follows by combining the estimates.

Reciprocal lower bound {#sec:lower_bound}
======================

This section is devoted to a proof of Theorem \[thm:main\]. We first state and prove the coarea inequality mentioned above which constitutes the main technical contribution of this paper. This corresponds to Proposition 15.7 in [@Raj:16], where a similar result is proved under the assumption that $X$ has the mass upper bound $\mathcal{H}^2(B(x,r)) \leq Cr^2$. The proof of Proposition 