To believe the world continues on after your death requires a leap of faith. After all, you have no evidence to suggest that the world is capable of existing without you sensing it; when you sleep, the world does not appear to exist as it does when you're awake. The state of the world is, sensibly, dependent on your state of awareness; its existence dependent on yours.
We have clear evidence of the world continuing to exist without us being aware of it, precisely because of what happens when we sleep, or are unconscious in some other way. If you sleep for 6 hours, when you return to consciousness your observations of existence are entirely consistent with the sort of observations you would have made had you been conscious of those six hours. Clocks have changed to indicate the passage of six hours. New posts and news stories, time-stamped to the missing six hours, will have appeared on the websites you frequent. The sun will have changed its position in the sky by the amount you would have expected it to in six hours. The programmes on TV would have changed to what was scheduled to have been on six hours after you went to sleep. The clothes you left to dry would have dried as much as they would have in six hours of constantly checking them. I could go on ad infinitum, but this all adds up to fairly good evidence that the world continues to exist in much the same way when we aren't conscious of it.
Interestingly, to believe that you continue on after your death requires no such leap of faith, but is heavily evidenced: for as long as you have been aware, you have been aware, and there is no time you are aware of during which you were not aware.
It's possible to be aware of times that you have not been aware, confusing as that sentence sounds. Aside from the evidence mentioned above that time passes without our noticing it, awareness is not a continual stream that flows without pause between periods of unconsciousness. I am often subjectively aware of an interruption and gap in this experience.
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