these continuity properties are fairly non-trivial. consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. then, however, the density is discontinuous, unless the given function is monotone. in other words, there is a conflict between good behavior of a function and good behavior of its local time. in this sense, the continuity of the local time of the wiener process is another manifestation of non-smoothness of the trajectory.