Continuity means consistency with the prior teaching, not absence of development.
Development that begins hundreds of years after the Apostles is not the Apostles teaching, nor continuous.
"Development" apart from Scripture is called false teaching.
The only thing continuous is paganism. Rome is the Borg of religions. Same philosophy too - Resistance is futile. You will be assimilated.
Come to think of it, Rome shares the same philosophy as Islam.