Let $(B_t)_{t\in\mathbb{R}}$ be a real stochastic process. $\forall\omega\in\Omega$, let $f_\omega(t) = B_t(\omega)$.
$\forall x\in\mathbb{R}$, $(B_t)_{t\in\mathbb{R}}$ is almost surely continuous at $x$ iff $f_\omega$ is continuous at $x$ for almost all $\forall\omega\in\Omega$.
Define sequential almost sure continuity at $x$ this way: $\forall (x_n)_n\subseteq\mathbb{R}\setminus\{x\}$, such that $x_n \rightarrow x$, it's almost sure that $B_{x_n} \rightarrow B_x$.
$\forall x\in\mathbb{R}$, is it true that sequential almost sure continuity at $x$ implies almost sure continuity at $x$?
If not, then also assume that, $(B_t)_{t\in\mathbb{R}}$ is almost surely continuous on $\mathbb{R}\setminus\{x\}$. That is, $f_\omega$ is continuous on $\mathbb{R}\setminus\{x\}$ for almost all $\forall\omega\in\Omega$. Would that be enough to show almost sure continuity at $x$?