Something along the lines of If A then B, if B then not. A has to be true before B can be true but if B is true first then A isn't true. Is there a term describing this in logic or philosophy?
The property you're looking for is called commutativity. We say a relation R is commutative if and only if aRb is logically equivalent to bRa. in a commutative relation order doesn't matter. In arithmetic addition is commutative because 2+3=3+2. In sentential Logic conjunction and disjunction and the biconditional are both commutative.
A relation is non-commutative if it isn't commutative. The material conditional is the only non-commutative binary relation in sentential logic. Division in arithmetic is noncommutative: 2/3 != 3/2
You have it switched, existence of a limit does not imply continuity (because it may not coincide with the value), but continuity does imply existence of a limit. Conditional statement with premise and conclusion switched is called the converse of the original. There is no special name either in mathematics or in logic for conditionals that are true without their converses, this is assumed by default, unless otherwise stated. On the contrary, conditionals that are true along with their converses are distinguished by terms "biconditional" or "logical equivalence". "Irreversible" or "non-reversible" may be used informally, but they are not terms.
As Conifold pointed out, there is no specific name for conditionals that are true without their converses in logic proper. However, in the theory of computation, there is a "reversible logic", where the term logic is used in the sense of logic gates.
The key consideration is that the relationship between the inputs and the outputs has to be injective (i.e. a one to one mapping), for this to be possible.