That continuity of function is arbitrary. In reality it provides people comfort in some idea of a soul but there's nothing suggesting it actually provides anything to the continuity of consciousness.
Between every loss in time, where you stop forming memories until you wake up again, you have nothing to affirm that your current consciousness is the same as your last waking period's. The only thing vaguely providing that illusion is your previously-formed memories, which would exist all the same on the digital mind, in theory.
What does maintaining continuity of consciousness look like to you? As in you are able to talk to your copy? And continue to live your normal life outside while your digital self lives their digital life?
Or are you saying you want the transition to digital to be seamless, where your digital self remembers laying in a chair, a quick pin-prick, and then they're in the digital realm?
Keep in mind, we have zero understanding of how you'd get the meat consciousness to transition into the digital consciousness - it's likely not even possible. The two options for copying are keep both alive or terminate the original somewhere before bringing the digital one online. There's many ways to do both, but those are the two.