The sense of continuity is held only by our memories. There is an analogy that illustrates this point. Scientists know that it takes a snail about three seconds to register light. So imagine that a snail was watching you, and that you left the room, robbed a bank, and came back in three seconds. As far as the snail was concerned, you never left the room. You could take her to court and she would provide a perfect alibi. For the snail, the time that you were gone from the room would fall into one of those gaps between the frames of flickering existence. Her sense of continuity, assuming snails have one, would simply not register the gap. So the sensory experience of all living beings is a purely artificial perceptual construct created in the imagination.
There is a Zen story in which two monks are looking at a flag that is waving in the wind. The first one says, "The flag is waving." The second one says, "No, the wind is moving." Their teacher comes over and they pose him the question. "Who's right? I say the flag is moving. He says the wind is moving." The teacher says, "You are both wrong. Only consciousness is moving." As consciousness moves, it imagines the world into existence.
So the mind is a field of energy and information. Every idea is also energy and information. You have imagined your physical body and the whole physical world into existence by perceiving energy soup as distinct physical entities. But where does the mind responsible for this imagination come from?