Yes, this seems almost certainly true (and I think is even necessary if you want to satisfy the VNM axioms, otherwise you violate the continuity axiom).

(and I think [a bounded utility function] is even necessary if you want to satisfy the VNM axioms, otherwise you violate the continuity axiom)

An unbounded function is one that can take arbitrarily large finite values, not necessarily one that actually evaluates to infinity somewhere.

Yes, this seems almost certainly true (and I think is even necessary if you want to satisfy the VNM axioms, otherwise you violate the continuity axiom).

An unbounded function is one that can take arbitrarily large

finitevalues, not necessarily one that actually evaluates to infinity somewhere.