Properties of Sample Functions of a Stationary Gaussian Process

Let $\{\xi_t(\omega),-\infty<t<\infty\}$ be a separable stationary Gaussian process with a continuous correlation function. Then, the following alternative holds true:
1) either for almost all w the sample functions of the process $\xi_t(\omega)$ are continuous functions of $t$.
2) or there exists a $\beta>0$ such that for almost all $\omega$ the sample function $\xi_t(\omega)$ is such that
$$\varlimsup_{t\to t_0}\xi_t(\omega)-\varliminf_{t\to t_0}\xi_t(\omega)\geq\beta$$
for any $t_0$.
In the second case almost all sample functions have no points of first order discontinuities.