Notice you can bound away from zero and this together with the unifrom continuity of said function allows you to pick a neighbourhood (in ) of any point in A on which there is a branch of logarithm defined. There may be some ambiguity over multiples of but the exponential doesn't see them, so you can ignore them and take the "true" extension. This proves A is both open and closed, and A is never empty because 0 is always in your set. I leave the formal proof to you.