A continuity criterion for a class of Markov processes

The following theorem is proved.
For a standard process $X$ with a standard adjoint process $\widehat X$, the conditions:
1) the sample paths $x_t(\omega)$ of the process $X$ are continuous a.s.,
2) $\forall\,f,g\in C_K\colon S_f\bigcap S_g=\varnothing$, $\langle P_t,f,g\rangle=o(t)$, $t\downarrow 0$, are equivalent.