There's nothing about the idea of physical consciousness that says it has to be a continuum -- there could just be some critical mass or qualitative attribute of brains that puts us "over the threshold", so to speak. Nobody can give any kind of a definitive answer. For ideas about a "continuum" of consciousness, you might read Phi:

http://www.amazon.com/Phi-A-Voyage-Brain-Soul-ebook/dp/B0078...

Or for other views, you might check out V.S Ramachandran (neuroscience): http://www.amazon.com/Brief-Tour-Human-Consciousness-Imposto...

Jeff Hawkins (computer science): http://www.amazon.com/On-Intelligence-Jeff-Hawkins/dp/080507...

Hofstadter (mathematics, cognitive science): http://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/...

Those are some of my favorite popular-press books on the subject.