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If I have a function that has no upper bound (i.e. is not bounded from above), is it acceptable to say that its upper bound is at infinity? If not, is there any way to describe that property, other than "not bounded from above"?

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Generally one says the function is "unbounded", although this has some ambiguity, as it could take on arbitrarily large positive values or arbitrarily large negative values. A slightly more natural way to phrase your statement is "not bounded above" rather than "not bounded from above".

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