That’s exactly the point: we want division to model inverses of multiplication, but an inverse requires a 1-1 relationship. What the vertical line tells us is

That’s exactly the point: we want division to model inverses of multiplication, but an inverse requires a 1-1 relationship. What the vertical line tells us is that division by zero can’t map to a unique value — it has to give us back the whole set. So we can discuss division by zero in the contexts of inverse maps, but there’s no way to have a function that inverts multiplication by zero. Small technical note: I think you mean ‘dx=0’.