When balancing on a knife-edge, it is dangerous to fall off. But consider whether it is more dangerous to stay on, since then you risk being cut in half. Falling off at least keeps you in one piece.
You only really get cut in half if you slide off in line with the blade, so sometimes it's not a choice between staying up or falling, but in choosing how you fall to minimise your loss.
Once fallen, one can climb to a more advantageous perch.
Stand on what you think is a stable surface, and watch slivers of it fall away one piece at a time, until you're left standing on one very thin, sharp piece.
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Once fallen, one can climb to a more advantageous perch.