I have been doing a bit of research on this topic lately: I have written off a few myths, and pondered on others (for excessive lengths of time).
From a physical approach, I once heard that the cells in the average human were all different to the cells that had been there approximately seven years ago. I soon discovered that this was not true about all cells, such as brain cells. (I suppose this is quite clear, because if we lost all of those over a period of seven years, surely we wouldn't remember anything.)
From a mental and emotional approach, we are always changing our minds et cetera, but fundamentally we think in the same way as we did seven years ago.
I eventually came to the conclusion that for this question to be answered thoroughly, one needed to define "the same person".
I thought about this carefully. Could "the same" mean that even if we lost a few skin cells or a bump on the head made us forget a few things we would still be effectively the same person? Well, I actually don't think so. Even if we are 99.999...% the same, we are only partially the same. We are partially the same as every other human on the planet, and we don't call ourselves the same person. I like to think that "the same" should mean "exactly the same", and therefore with every instant (or frame, like in a film) we are somehow different than before. Whether it be that we are thinking different things, or that we have lost a couple of dead cells, or that our temperature has changed... anything.
And there you have it. At least, that's my view.