= Solution
Use the standard theorem that supercritical Bernoulli percolation on $\mathbb Z^3$ has a unique infinite open cluster almost surely. Fix $p>p_c$ and choose $q\in(p_c,p)$. Almost surely $\eta_q$ has an infinite cluster somewhere. On $I_p$, the origin and that cluster lie in the unique infinite $\eta_p$-cluster, so a finite $\eta_p$-open path joins them. Almost surely every label on this finite path is strictly below $p$; increasing $q$ to some $r<p$ above those finitely many labels makes the origin percolate in $\eta_r$. Hence $I_p\subseteq\{M<p\}$ up to a null event, and part e gives left continuity at $p$. Together with part c, $\theta$ is continuous on $(p_c,1]$.
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