Your question begs the preliminary question, what is the self?
Modern neuroscience supports the idea of it being a mental construct produced by the brain as a kind of coat rack for all the things going on within it and associated with your physical senses. Clearly, the propensity to build such a model is genetically encoded and unavoidable. However some meditation adepts claim to be able to bring it to a temporary standstill, and some may refer to this mental state as "nirvana" or "enlightenment".
Much high-level neural processing can take place subconsciously. Indeed, many if not all thoughts which rise to consciousness were in fact subconsciously decided upon perhaps a second or so previously; for example the intention to press a button may arise around that length of time before you become aware of your decision. However a practiced reflex action, say a racing driver changing gear, may happen considerably faster. Your brain plays tricks on itself to synchronise it all into a seamless whole (much as a digital video player must synchronise its visual and audio output signals following processing at different speeds).
What seems to be happening is that the skimmings of this subconscious life get injected into a certain area of the brain which, uniquely, generates the qualia of conscious experience. Thus, the self we experience (and instinctively believe in) is just a summary model of the "true" self maintained by the subconscious brain.
The experience of continuity is an artefact of the brain's model (even the hardware divides into into short-, medium- and long-term memory architectures), which enables it to sort memories into temporal order and thus build a useful model of past and present events. We only feel the same person each morning because the brain reloads sufficient memories to make sense of waking up, much like logging back in to your user account at work and seeing the familiar desktop.
Now, you ask, when this lot gets shut down and rebooted, perhaps even cloned, is it the "same" self? I hope you can see by now that the "atomic" self we experience and instinctively believe in is over-simplified, and clinging to it is unhelpful. The roots of your self are the mental model and associated memories. Boot those on any suitably architected platform, be it hard- or wetware, a clone perhaps, and it will feel like the same old you. But any clone's experiences and hence recent memories will immediately diverge to become patently different selves. Kill off the ones with the longest memories and you have killed off your old self, but you can let them live if you want to. Either way, you are still your new self.
So there it is; the atomic and continuous self you agonise over is a delusion; It vanishes when you go to sleep and gets rebooted afresh for every dream and every morning. The self which underpins it is contingent on its past and survives the night. A clone may be regarded initially as a new "instance" of the saved self, but immediately diverges as a new one. The relationship of the self to those elusive qualities of conscious experience remains the hardest problem in the philosophy of mind: all bets are off.