I recall as a young child that I didn't quite recognize that the "me" of tomorrow and the "me" of yesterday was quite the same as myself today. Once a punishment for a misdeed got delayed until tomorrow, which made me very happy; the next day the mere suggestion of implementing the punishment seemed grossly unfair to me: I didn't do it! it was done yesterday! what does it have to do with me today?!
I think other kids have this waking-time-limited self awareness: they are generally bad at accepting delayed gratification beyond the same day, and they sometimes dislike going to sleep to the extent that seems almost similar to the fear of death for grown-ups: the self of today disappears, and what does it matter that another being tomorrow would inherit most of the self-of-today memories?
Thus the first question to those more versed in psychology than I am: does the above make any sense?
Sometimes I think that there is a time span of self-awareness for grown-ups as well. A teenager often doesn't consider the 30-something that he will become as someone worth too much consideration; self-of-this year is what matters most. A 20-something, and often 30-something pays little attention to the retired self of 70-something. It seems that time span of self awareness has increased, from one day to a few months to a few years, but for most people didn't quite disappear.
Thus the second question: is there any truth to the above?
Next, I would like to look at the above from the viewpoint of philosophy rather than psychology. Does it make sense to consider a self not as one person, but as a distribution of self-awarenesses that varies with time?