= Solution
Work on the common <probability space> from part (c). For a configuration of all the uniforms, the <set> $S=\{r\in[0,1]:I_r\text{ occurs}\}$ is an upper interval, possibly with or without its lower endpoint. It is nonempty because $I_1$ occurs, and the <coupled onset parameter for origin percolation> is $M=\inf S$. For $0<p\leq1$,
$$
\bigcup_{\substack{r<p\\r\in\mathbb Q\cap[0,1]}} I_r=\{M<p\}.
$$
Indeed, occurrence at some $r<p$ implies $M<p$. Conversely, if $M<p$, the definition of infimum yields occurrence at some $s<p$, and a rational $r$ between $s$ and $p$ also satisfies $I_r$. This also proves measurability of $M$ through its strict sublevel <sets>.
Choose a rational sequence increasing to $p$. By <continuity from below of a measure> and the <monotone coupling of Bernoulli percolation>,
$$
\lim_{r\uparrow p}\theta(r)=\mathbb P(M<p).
$$
Moreover, $\{M<p\}\subseteq I_p\subseteq\{M\leq p\}$. Hence $I_p$ is the disjoint union of $\{M<p\}$ and $I_p\cap\{M=p\}$, giving
$$
\boxed{\lim_{r\uparrow p}\theta(r)=\theta(p)-\mathbb P(I_p\cap\{M=p\})}.
$$
It is important to retain the intersection with $I_p$: occurrence at the infimum is not automatic. At $p=0$ there is no left limit within the parameter domain, so the displayed left-limit assertion is for $p>0$.
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