Before we start, I'm not talking about the continuously growing teeth of rodents, or the tusks of elephants. Instead, what I'm looking for is one set of teeth that could fit in a newborn's skull, that grow in diameter and length as the animal matures.These teeth would not be replaced with larger teeth like the typical mammalian route of milk teeth to permanent teeth; the teeth you are born with are the teeth you keep.
I'm thinking their growth would be more analogous to the lengthening of bone, but I'm not sure applying the same process would yield adequate results. I've also considered them growing from periods of shedding, like the scales of reptiles, which may involve the animal needing to chew tough fibrous foods to slough the 'dead' layer off. I've thought of a mechanism similar to how trees grow in diameter, but I don't know if that would work either. Lastly, I've considered teeth like parrotfish beaks, where the tooth is just comprised of several very small teeth, but I've already seen someone do this and I don't wanna be a copycat lol.
What I need to know is if any of these solutions (or some external mechanism) could produce what I want given the following stipulations:
The teeth have to have a genetically inherited and consistent shape. In other words, if the tooth in one specimen is triangular with ridges along the edge, then that trait should be seen across all members of it's species.*
The tooth needs to have a similar rigidity and sharpness to most earth animals. They cannot break constantly or deform under regular use.
The tooth has to be vascularized and innervated in some way; so something similar to a root might be necessary.
The tooth cannot have a 'cap' like seen in the horns of earth animals.
The tooth should be made out of materials found readily in an earth-like ecosystem.
The tooth should not dehydrate easily.
The tooth MUST grow with the animal; keeping the skull it's newborn size is not an option.
*The tooth could be one particular shape genetically, but is widdled down to another (still effective) shape with regular wear, like the self sharpening molars of Triceratops.
(If anyone is wondering why these stipulations are in place, I'm designing an alien that nurses via maternal dermatophagy, and they cannot afford to be without teeth in any part of their lifecycle)