From a traditional Halakhic perspective, especially a historical one, is the child of a Jewish father and a mother of unknown status, for whatever reason undisclosed (I.e. analogous to when undisclosed by the mother), by the father, considered Jewish?

I understand a situation where the mother is unknown is… logistically difficult to arise, but in this context, let’s assume the child was born in an unknown place and later brought by a Jewish father to his own community. The child is not mamzer regardless of the mother’s status. So, presuming two situations:

1) The child’s father discloses that the child’s mother is Jewish. Is the father to be taken at his word for it?

2) The father for whatever reason doesn’t disclose. Is the child’s status going to be reliant on the area in which the child was born? And if, in the example above, the child was born in a completely unknown area to the local community, and the father keeps it a secret, is the child to be considered Jewish or potentially a gentile?

I couldn’t find a response to this because, well, it’s a very unlikely situation.