Change and transition

It seems that necessarily:

1. If an object was F at t₁ and non-F at t₂, and the object existed at both of these times and all intermediate times, then the object changed with respect to F.

The at-at theory of change adds that this just is what change is, but in any case it seems quite uncontroversial that being F at one time but not another, while persisting from one time to the other, is sufficient for change.

But suppose that a lamp is on at all times that are rational numbers and is off at all times that are irrational numbers. Then according to (1), the lamp changed with respect to being on. Yet when did it change? A change is always a transition from one state to another. Suppose I take an ordinary lamp that's turned on at noon, off at 1, on at 2, and off again at 3. Then we can say that it changed from off before noon to on at noon; from on before 1 to off at 1, and so on. But there was no transition from on at noon to off at 3--there was, instead, a series of transitions. Likewise, then, if the lamp is on at t₁ (which then is rational) and off at t₂ (which is rational), it didn't transition from on at t₁ to off at at t₂. For after t₁ and yet before t₂, then lamp was off and then on and then off and so on, infinitely many times. So just as the ordinary lamp didn't transition from off at noon to on at 3, the weird lamp didn't transition from on at t₁ to off at t₂. Hence, it never transitioned.

What should we say? I guess we have three options. First, deny the possibility of a dense time sequence. Second, deny (1). Third, deny that change requires transitions.