This is a powerful question and i give major kudos for thinking it up. I don't think there's any way to know, if anyone can determine this they are worthy of the nobel prize. Be careful, you could be cast into the nut house for simply wrapping your mind around this one.

Some two thousand years ago, Heraklit said: you never go in the same river twice. It is this same thought: all the water is continuously exchanged, the water line is in permanent change, sand banks move, shift, appear, disappear. Yet the same river. It is not the water, it is not the water line that defines the identity of the river. It is its continuous existance, its being the same from every change to every change. I'm definitely not the "same" person I was twenty years ago, that would be an outrage. I'm the successor to my history, the heir to all my memories (fake or real) , and I carry on. We are more of a tradition, so to say...

I've actually thought about this question quite a bit just within the past few months. It's a very interesting topic. Unfortunately, in my opinion the question does not really have an answer. As the question is posed, it does not define what "continuity of consciousness" means, nor does it specify or suggest any possible form of proof or disproof. There is no obvious experiment one could perform on oneself or on another person to detect "continuity of consciousness." Any form of memory test would be irrelevant, as clearly stated in the question. How would we know continuity of conscience if we saw it? If a computer program were specified to have continuity of consciousness, could we test the program in order to verify that the specification had been met? We could perform such a test only if we were also given a criterion for success. Once we know what "continuity of consciousness" really means, I think we can try to answer this question. For now, I don't think an answer is possible.

In my opinion, your sense of a constant self/identity is an illusion that your brain forces upon you in order for you (i.e. your body) to function better in the world. This helps preserve your body's existence over time by making you want to take steps to ensure your (illusory) self's continued existence. For example, steps like eating, not wandering off cliffs, saving money for future needs etc. If you didn't have this illusion that YOU were also gonna be here later, then why do all that life-preserving stuff? In reality though, I don't think there is a continuous person/consciousness/identity over time. Sure, in a minute there may be a guy who feels exactly like I do right now, but he has no (CAN have no) strong connection to the "present me", such that he would feel different if the "present me" were killed and he (a copy) was magically created out of thin air as a replacement. He would be no more or less "present me" in either case. The "future you" is the same as the "present you" only in the sense that e.g. a webpage in your browser is the same page as the one on the server from where you fetched it.

It can't be known. Our consciousness could change bodies every second and as long as it doesn't retain anything from the previous brain, not any memories or any aspect of the person's personality or thoughts and upon switching was instantly aware of all of the new body's memories, personality, and thoughts then it would seem exactly the same as though our consciousness just stayed in the same body forever. The experience in that second, including your experience of what you think happened beforehand would be exactly the same. Or each consciousness could die every five minutes replaced by a new one and as long as they both have the same access to the brain we'd have no way of knowing.

Given that we don't even have a very good understanding of what consciousness means, this is a pretty complicated question. If consciousness is a process and not a state, then no, your identity is not an illusion. Obviously many things about an individual change over time, including cognitive aspects of the individual, but unless you suffer a serious brain injury that disrupts your memories, attitudes, beliefs, values, and personality, then the salient features of your identity preserve "you" even as they gradually change through time. I think it is the myriad relationships in your mind that make you who you are, and it would take a catastrophe to suddently and simultaneously change enough mental relationships to make you a different person. Think of a vortex of water, like when you're draining a bath. At any given moment, the water molecules that make the vortex are not the ones that were there a moment ago, but the form persists because of specific physical relationships that obtain throughout.

It's not an illusion, it's a fact. We are the sum of our experiences and actions, not a mere moment in time or accidental coincidence of the timeline. Our soul is continuous, our body is born and dies. What we do during that time determines our soul's fate in whatever afterlife you hope to believe or refuse to accept.

I don't mean this to sound flippant, please don't think it is because I actually think it is a good question that deserves a good answer. However, I think the only honest answer that we can give to this is: who knows? There is certainly, at the very least, the illusion of continuity and maybe that's all that really matters. It maybe impossible to ever accurately answer this question, so perhaps we have to be satisfied with the effect of continuity of consciousness. Or perhaps the answer is kind of both and neither. If we accept the premise that consciousness, and specifically identity, is the sum of our experiences combined with inate psychology and neurophysiology then it maybe that identity is constantly changing and yet always the same thing. Like a lump of wet clay. Everytime you touch it or put pressure on it it changes and yet it is the same lump of clay.

I'll use a rivulet example to explain how I think of this. Let's say your consciousness is a rivulet of some substance running down a tree. The front end of the rivulet, as it makes its way down the tree, is current experience. Everytime you go to sleep or get knocked out, the rivulet merely stops. It still has the rest of the rivulet (what brought it there) behind it, or further up the tree, it's just not currently flowing. There are other substances (other people) flowing down the tree as well, and those can shape your path, as well can the bark (outside experiences). This way of viewing things answers a number of questions: Continuity of consciousness is not an illusion. Where we are now on the tree is a product of which way the rivulet went before, and we can still see it further up the tree...it's still "us" and it doesn't matter if out current state stopped for a while (sleep/temporary unconsciousness). Also, this explains the fact that if you clone yourself (create a rivulet of the same substance), you don't achieve continuity of consciousness. The only way to achieve eternal consciousness is to upload the brain onto a non-organic substrate (substrate independence for the win!). I know it all sounds a little Chinese poem-ish, but I hope this makes a bit of sense.