I'm not sure what your argument really is here. Your very invocation of "the wave function" implies continuity in the evolution of the state vectors, described

I'm not sure what your argument really is here. Your very invocation of "the wave function" implies continuity in the evolution of the state vectors, described by the wave equation - it matters not that results of macroscopic measurements will only produce quantized results. You still used continuity of the more abstract (but no less "real") state vector evolution.