You have some assumption wrt time in there. To say that your consciousness is continuous relies on you trusting your memory of it being so, which, lacking a time machine, is no more provable or observable than others consciousness.
Removing the continuity assumption, perhaps even time, kind resolves the duplicate question: both copies (if conscious) should have a memory of conscious you and experience their consciousness being a continuation of that, while you, now you, are as removed from those future copies, as you are from me.
One question, that seems tied to consciousness, is what is "now"? Are there objective reasons to believe in the existence of a canonical "now" existing independent of conscious experience? If not then there really is no point in distinguishing either momental or continuous experiences as special in any way. Perhaps consciousness just emerges as an interpretation of a single state connecting past and future by memory and anticipation, it's simply appears continuous like the series of images at the top of the article appears to be.
Yes this is my favorite explanation: that consciousness is incredibly brief and delicate. That only one very precise organization of the brain could give rise to the epiphenomenon that is me. So precise in fact that "I" only exist for a very brief moment.
What I perceive as a continuous consciousness is really just a memory of other "I"s who were caused by this brain in the past, left a memory of introspection and then dissipated to be replaced by an new one.
This view has many advantages: it solves the problem of time you elude to, it answers the problem of other minds and the problem of my childhood self. The only problem I can not see solved is the hypothetical molecular level duplicate I mentioned above.
Removing the continuity assumption, perhaps even time, kind resolves the duplicate question: both copies (if conscious) should have a memory of conscious you and experience their consciousness being a continuation of that, while you, now you, are as removed from those future copies, as you are from me.
One question, that seems tied to consciousness, is what is "now"? Are there objective reasons to believe in the existence of a canonical "now" existing independent of conscious experience? If not then there really is no point in distinguishing either momental or continuous experiences as special in any way. Perhaps consciousness just emerges as an interpretation of a single state connecting past and future by memory and anticipation, it's simply appears continuous like the series of images at the top of the article appears to be.