You raise an interesting point, to which I would respond this way: the purpose of the axiom called local non-satiation is to provide as weak as possible a requirement on preferences that, when combined with the continuity axiom, will (ultimately) ensure the consumer's chosen bundle lies on rather than inside the budget constraint. This these two axioms do by ensuring that IF a best bundle exists on the budget set THEN it must be somewhere on the boundary of that budget set, not in the interior.

Preferences such as, say, u(x,y)= -(x+y) are indeed ruled out by the axiom: As you note, below, with these preferences, while it is almost always possible to find a better bundle in the consumption set, it is not ALWAYS so---in particular, it is not possible at the origin. However, if you were to define preferences such that, say, u(x,y)= -(x+y) for all (x,y) not equal to (0,0), and u(0,0)= -1, then strictly speaking these would satisfy the axiom of local non-satiation--though not the axiom of continuity.


Does that help?