= Solution
Let $A_n$ be the event that the origin has an open <graph path> to the boundary of $[-n,n]^d\cap\mathbb Z^d$. The <graph path> can be stopped on its first boundary visit, so $A_n$ depends on only finitely many sites. Its <probability> $f_n(p)=\mathbb P_p(A_n)$ is a finite sum of terms $p^a(1-p)^b$, and is therefore a <continuous function> of $p$. Also $A_{n+1}\subseteq A_n$, and the <König infinity lemma> gives $\bigcap_n A_n=\{0\leftrightarrow\infty\}$. By <continuity from above of a measure>,
$$
\theta(p)=\inf_{n\geq1}f_n(p).
$$
The <monotone coupling of Bernoulli percolation> described below shows that $\theta$ is nondecreasing. For $p<1$, its right <limit of a function> $\theta(p+)$ exists and is at least $\theta(p)$. For every fixed $n$,
$$
\theta(p+)\leq\lim_{r\downarrow p}f_n(r)=f_n(p).
$$
Taking the infimum in $n$ proves \b[$\theta(p+)=\theta(p)$]. This proves <right continuity of percolation probability> throughout $[0,1)$; at $1$ right continuity is understood relative to the parameter domain. The argument does not interchange two uncontrolled limits.
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