Although I argue that a time traveler who prevents his own birth does not cease to exist, time travelers would nevertheless need to proceed with extreme caution. How can they know which kind of scenario they are in? -
(i) you cannot prevent your birth because your timeline is single and continuous;
(ii) you can prevent your birth because your timeline is single but discontinuous;
(iii) you can prevent your birth because timelines are multiple and divergent.
If a group of people found themselves in what seemed to be their own California long ago and if they wanted to return to their remembered version of the twentieth/twenty first century, then they would have to be very careful not to change the course of events in any way. Rolfe tests whether the Gate has taken him to the past:
"'The first time I came through, I carved numbers on rocks in places I could locate on both sides - boulders, cliff faces - carved them deep enough to last for thousands of years. There's no trace of them back on our side of the Gate, where we know it's 1946. I'm still going to get some astronomers to look at pictures of the night sky - the stars change with time, you know - but I'm pretty certain this is the same time as back in California, the spring of 1946.'" (pp. 26-27)
So he is in the present of an alternative timeline, not in the past of his original timeline although, even here, he was taking a risk. He might have been changing the past/initiating a divergent timeline by carving on those rocks.
Poul Anderson's The Corridors Of Time and There Will Be Time are set in scenario (i). His Time Patrol series is ambiguous between (ii) and (iii).