I need to prove or disprove that for a stochastic process $(X_t)_{t \in [0,1]}$ with marginals $(\mu_t)_{t \in [0,1]}$ on $\mathbb{R}$, if the sample paths of $(X_t)_{t \in [0,1]}$ are continuous, then $(\mu_t)_{t \in [0,1]}$ is weakly continuous in $t$. I see a similar question [Continuity of the densities of a stochastic process](https://mathoverflow.net/questions/370159/continuity-of-the-densities-of-a-stochastic-process?noredirect=1&lq=1&newreg=56800865f773464bb9d45240cac0863a), but it's for the discontinuity of densities of $(X_t)_{t \in [0,1]}$. Any help is greatly appreciated.