My intuition as well. Continuity seems less of a big deal when we imagine computer hardware intelligence scenarios.

As another scenario, imagine a computer based on light waves alone; it's hard to see how a temporary blocking of the input light wave, for example, could cause anything as substantial as the end of a conscious entity.

However, if I think too much about light waves and computers, I'm reminded of the LED cellular-automaton computationalist thought experiment and start to have nagging doubts about computer consciousness.