The definition for a function being continuous at a point is obviously well known: Given there exists such that
But assume we use the following "definition"; Given there exists such that .
Where does this fail in implying continuity, how is this insufficient to showing that the implication:if , then holds (which is the sequential definition of continuity)?
Well, the only method I could see working is that if that definition worked, then it's negation must always be false, that: given , there exists such that and must always be false. If I can find a case where the negation is true, then obviously the negation is not always false and therefore the new definition cannot imply continuity. But that is as far as I have gotten so far.
Or I guess the new definition just shows that if , then but then I'd just have to show that that is not equivalent to if , then but I do not know how to go about that exactly.