Is it possible for a function to be continuous at all points in its domain and also have a one-sided limit equal to +infinite at some point?
Yes, it is possible. (But the point at which the limit is infinite cannot be in the domain of the function.)
This requires three things:
Note that this implies that the limit is finite. (Saying that a limit is infinite is a way of explaining why the limit does not exist.)
3) items 1 and 2 are the same.
Relating to item 1 recall that
So, if the function is to be continuous on its domain, then all of its limits as
We can make one of the limits
Once you see one example, it's fairly straightforward to find others.
Is continuous on its domain, but
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