Say I go back in time. Does it mean I disappear from the time I was in? And I'm in two places at a point in the past (the me from that time and the me from the future)? Now if I want to go back to the time I came from, do I join back at the exact time I left (which means no one would notice my travel), or would I have to travel back to the time I left + the elapsed time I spent in the past? And if I really did return to the exact time I left, does that really mean no one noticed my leave? Because I'm sure when I left time continued and somebody noticed.
In the form of time travel being talked about, the answers are:
a) Yes, you disappear from the time you were in.
b) Yes, you're in two places at once for the duration between your arrival and your departure
c) Forwards travel in time doesn't cause too many philosophical problems (since we're already doing it) which is why nobody talks about it. But there's no requirement to wind up exactly where you started.