Probability stochastic process for all points of time

Question

Let $\{X(t):t\in[0,\infty)\}$ be a stochastic process and $f:[0,\infty)\to\mathbb{R}$ be a function. If the probability $$ \mathbb{P}(X(t)\leq f(t))$$ is known for all (fixed) $t\in[0,\infty)$, then, what can be said about the probability $$ \mathbb{P}(X(t)\leq f(t)~~\forall t\geq0)?$$