If a stochastic process is path-wise continuous then its filtration is right-continuous? Let $(\Omega,\mathcal{F})$ be a measurable space, carrying a stochastic process $X=X_{t≥0}$ with state space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $\mathcal{F}_t = \sigma(X_s:s≤t)$. Assume that trajectories $t\mapsto X_t(w)$ are continuous for all $w\in\Omega$. Prove or disprove : $(\mathcal{F}_t)$ is right continuous.

The filtration $(\mathcal F_t)_{t\geqslant0}$ may be discontinuous. For an example, assume that $X_t=t\cdot U$ for some random variable $U$. Then $t\mapsto X_t$ is almost surely continuous, $\mathcal F_t=\sigma(U)$ for every $t\gt0$, and $\mathcal F_0=\{\varnothing,\Omega)$, hence $\mathcal F_{0+}\ne\mathcal F_0$ as soon as $U$ is not almost surely constant.
The filtration $(\mathcal F_t)_{t\geqslant0}$ may be continuous. For an example, assume that $X_t=\mathrm e^t\cdot U$ for some random variable $U$. Then $t\mapsto X_t$ is almost surely continuous, $\mathcal F_0=\mathcal F_t=\sigma(U)$ for every $t\gt0$, hence $\mathcal F_{0+}=\mathcal F_0$.
