What about ? It is , it goes to ∞ at x=0, it's integrable on (0,1] (by comparison with ), and it's integrable on [1,∞) (by comparison with ).
Is there a function of class defined on such that and exists and is finite?
I found a function:
which fits everything except it's only class (because DNE).
Any ideas? I'm sure such a function exists; I'm just not sure whether it can be expressed in terms of elementary functions.
This is not a homework problem; I was just thinking.