There is a continuation, but there is no continuation of consciousness, which completely breaks down at the death of the individual.
Except that in the Abidhamma account of things, there isn't some fixed entity known as "consciousness" which breaks down at the moment of death; rather, as I understand it least, there's a succession of cittas (consciousness moments), each conditioning the next. And that process continues across lives via the paṭisandhi-citta ("relinking citta").
So it is a type of continuity, in the sense that there is continuity across a chain or continuum.
The Suttas and Abhidhamma slightly differ on this account. At an individual's death, according to the Suttas, consciousness (viññāṇa-kkhanda) breaks down like the other aggregates.
Although citta is said to be synonymous with viññāṇa, in Nyanatiloka Thera's Buddhist Dictionary
, viññāṇa is translated as consciousness, whereas citta is translated as mind, consciousness and state of consciousness. I prefer to use the term mind for citta, as to differentiate the Sutta-based and Abhidhamma-based models.
For bhavanga-citta, it is translated as subconscious and 'life-continuum.
So the Suttas state that all five aggregates end at death, but the Abhidhamma states that there remains a form of consciousness or mentality. I think this conundrum is due to terms. For there to be karma-formations (saṇkhāra), some form of mentality is probably needed.
Here is XIV of the Visuddhimagga:
“As soon as rebirth-consciousness [paṭisandhi] (in the embryo at the time of conception) has ceased, there arises a similar subconsciousness with exactly the same object, following immediately upon rebirth-consciousness and being the result of this or that karma (volitional action done in a former birth and remembered there at the moment before death). And again a further similar state of subconsciousness arises. Now, as long as no other consciousness arises to interrupt the continuity of the life-stream, so long the life-stream, like the flow of a river, rises in the same way again and again, even during dreamless sleep and at other times. In this way one has to understand the continuous arising of those states of consciousness in the life-stream.”