# a.s discontinuous on an interval but continuous in Probability?

Can a Gaussian process be almost surely discontinuous on an interval T but at the same time be continuous in Probability everywhere on T?

Alternative question: can a sequence of discontinuity points converge to a point where the Gaussian process is continuous?

Thank you very much for your input!

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Robert J. Adler's book An introduction to continuity, extrema, and related topics for general Gaussian processes has a lot of examples like this. In particular, on page 116 (exercise 6.1) he notes that if $(X_t)$ is Gaussian, and stationary with $\mathbb{E}|X_u-X_0|^2\sim 1/|\log(u)|$ the sample paths are discontinuous, though the process is continuous in probability.