= Solution
Use the following general result, including its site version: independent <site percolation> on $\mathbb Z^d$ has <almost surely> at most one <infinite percolation cluster> at each fixed parameter; for every fixed $r>p_c$ it has exactly one <almost surely>. This is the <uniqueness of the infinite percolation cluster>, also called the <Burton-Keane theorem>. For $d\geq2$, the additional general fact $p_c<1$ ensures a choice of a parameter strictly between $p_c$ and $1$. In dimension one, $p_c=1$ and the asserted interval is empty.
Fix $p\in(p_c,1]$ and choose $q\in(p_c,p)$. Let $\mathcal C_q$ be the unique infinite open <connected component of a graph> at $q$. Under the <monotone coupling of Bernoulli percolation>, it is contained in the unique infinite open <connected component of a graph> at $p$. On $I_p$, the origin and $\mathcal C_q$ therefore lie in that same <connected component of a graph>, and a finite $p$-open <graph path> connects $0$ to some <graph vertex> of $\mathcal C_q$.
For this fixed deterministic $p$, the countable <set> of uniforms satisfies $U_v\ne p$ at every <graph vertex> <almost surely>. Thus all uniforms on that finite <graph path> are strictly less than $p$, including its endpoints. If $t$ is their maximum, choose $r$ with $\max(q,t)<r<p$. The <graph path> is then $r$-open, and its endpoint remains connected to infinity through $\mathcal C_q$. Consequently $I_r$ occurs and $M<p$. We have proved
$$
\mathbb P(I_p\cap\{M=p\})=0.
$$
Part (d) gives left continuity at this $p$; part (b) supplies right continuity when $p<1$. The same finite-path proof gives left continuity at $p=1$, since all uniforms are strictly less than $1$ <almost surely>. \b[Thus $\theta$ is continuous on $(p_c,1]$], without asserting continuity at $p_c$. This is <supercritical continuity of percolation probability>.
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