I would expect this question to be answered somewhere but I can't find it. I have the following conjecture.
Let $X=(X_t)_{t\geq0}$ be a time homogeneous (strong?) Markov process with cadlag paths on some general state space (e.g. Polish metric space) and let $s >0$. Then for all starting points $x\in S$ $$ P_x(X_s \neq X_{s-}) = 0, $$ where $X_{s-} =\lim_{t\uparrow s} X_t$ denotes the left limit point $X$ at time $s$. In other words, $X$ is almost surely continuous at any fixed time.
I strongly believe that there is a contradiction to time homogeneity if we assume that there is a time point $s>0$ with strictly positive jump probability but I have been unable to construct such a contradiction.
I would be very thankful for any hints on a proof or pointers to relevant literature.