I'd go deeper and say that “continuous consciousness” isn't a concept that makes sense. You only live in the moment, with access to part of your past selves’ memories.
I posit that consciousness is a chemical process occurring in your brain. This process is continuously ongoing, and when it stops, you die. If a transporter constructs a perfect copy of you, down to the chemical process that constitutes your consciousness, then there is no continuity between your original body and this new one, because it's a wholly different brain.