What constitutes you. Every day some of your cells die and others are created to replace them, some memories are lost (perhaps only temporarily) and new memories are formed, and over time our neuroplasticity shapes the very physiology of our brains. Where is the you that you identify as a constant through all this change?
When I wake up I often have a few seconds of coherent intelligent thought before I remember what my life is like, where, and who I am. During these few seconds I sometimes wonder about whether I am young or old, rich or poor, and I can revel in the simple state of being where I am unaware of pretty much everything. A state where all my thoughts are occurring for the first time, devoid of context.
A similar situation is invoked as a plot device in Dark City, which is a shit film (much worse than its IMDB rating suggests) but it does leave a lasting impression. Each night, the memories and lives of all the inhabitants are swapped around and they all live the next day with a new husband/wife, a new job, new hobbies etc. and they are unaware of their previous/original lives.
The classic example of a thought experiment about continuity of self involves a star-trek transporter. This device maps out all the atoms in your body and then simultaneously destroys the original and creates a duplicate elsewhere. Is the transported you you? What if the original weren't destroyed - which one would be you, or would both be?
The most common real life application of this concept is in drugs, especially ones that change personality. If I took a pill that gave me empathy and turned me into a sensitive metrosexual man then I would not be me. Whoever that person was may fit in better with society, and they may be happier than I am, but they would not be me.
I think we are more accepting of gradual changes in our self - if we go to therapy or just decide to work on an aspect of ourselves then we are changing our thought patterns; that can lead, slowly, to physical changes in the brain. If it happens overnight then we are much more wary of it, but where is the logical difference?
You may find this interesting.