A random function of a variable t in T (or, a random function on T) is known as a function f whose values are random variables all defined on a common probability space, where T is an arbitrary set. A random function is also called a stochastic (random) process. In this work, we base ourselves on a random function of E-process type and such a function is also called a random function, briefly. In this approach, the domain of such a random function is R or an interval of R, and the set of values of this random function is considered as a special probabilistic metric (PM) space (more precisely, an E-space) of metric space-valued random variables, and all our definitions and results are presented using the tools of PM spaces. In this context, we introduce the concept of an exhaustive family of such random functions, which is a natural generalization of equicontinuity, and we investigate its basic properties. We also examine some of the properties related to the continuous convergence in probability for a sequence of such random functions and certain conditions which give rise to the continuity in probability of the limit of a sequence of such random functions.