Consider a random field on $R^d, d \geq 1$. A simple condition is given on the covariance function which ensures the existence of a version of the random field in which the realizations are everywhere continuous. The proof involves a rather delicate approximation of the random field by interpolating polynomials of suitably high order.
John T. Kent. "Continuity Properties for Random Fields." Ann. Probab. 17 (4) 1432 - 1440, October, 1989. https://doi.org/10.1214/aop/1176991163