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Decius comments on On the importance of taking limits: Infinite Spheres of Utility - Less Wrong
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No, I mean a function whose limit doesn't equal its defined value at infinity. As a trivial example, I could define a utility function to be 1 for all real numbers in [-inf,+inf) and 0 for +inf. The function could never actually be evaluated at infinity, so I'm not sure what it would mean, but I couldn't claim that the limit was giving me the "correct" answer.
Um... -inf and +inf are not real numbers. (Noting that your function as described is undefined at -inf.)
In addition, the definition of continuous restricts it to points which exist on an open interval; if the limit from below and limit from above are equal to the value at X, then the function is continuous on an open interval containing X. How do you determine the limit as X approaches +inf from above?
MrMind explains in better language below.