You cannot witness the persistence of the 'I'-self in time since it is only located in the 'now' and there cannot be two different 'now's at once. You can witness all kinds of feelings and thoughts, but only in the 'now' - everything else is a thought. That you persisting in time, that the same self recurs and in that recursion makes meaning is a large part of thought. It's not really essential to anything but that is where it starts and ends - it acts like a filter that smooths the course of experience by blurring it, more or less.