It is shown in this paper that the paths of a real separable Gaussian process on an interval have a number of interesting properties with probability zero on one. These properties include boundedness, bounded variation, continuity, uniform continuity, absolute continuity, a Lipschitz property, differentiability, absence of oscillatory discontinuities.
"Some Zero-One Laws for Gaussian Processes." Ann. Probab. 1 (2) 304 - 312, April, 1973. https://doi.org/10.1214/aop/1176996982