Consider a Markov process X_t with state space E, that is a Borel set of a compact metric space, which admits a local time L^x_t at all points x in E. Finding necessary and sufficient conditions for the joint continuity in (x,t) of the local time, is a problem that has attracted many researchers in the field during the past few decades. Interesting partial results exist by Marcus and Rosen and by Barlow, which hint towards an intriguing connection between the continuity of local times of Markov processes and the continuity of paths of an associated Gaussain process. We study the continuity of local times of Markov processes that have a strong dual (a quite general class that contain the processes treated by Marcus and Rosen and by Barlow). We explain through a functional Central Limit Theorem the above connection and find a necessary and sufficient condition for the continuity of the local time and for the tightness in C(K) of sequences satisfying the CLT (with K and compact set contained in E). This is a joint work with Nathalie Eisenbaum from CNRS Paris VI