Is the prophetic lineage historically continuous?

Fatimah's children, may Allah be pleased with her, namely Al-Hasan and Al-Husayn and their descendants, are attributed to the Prophet, may Allah's prayers and peace be upon him, as an honor to his lineage. Not everyone who claims kinship with the Ahl al-Bayt is truthful, and it is necessary to verify the correct and well-known lineage among genealogists. Al-Hasan ibn Ali, may Allah be pleased with him, was born in the 3rd year of Hijra and passed away in the 49th, 50th, or 51st year of Hijra. Al-Husayn, may Allah be pleased with him, was born in the 4th year of Hijra and was martyred in the 61st year of Hijra. Both Al-Hasan and Al-Husayn had sons and daughters, and the lineage of some of them continues to this day. Al-Hasan had descendants through his two sons, Al-Hasan and Zayd, and Al-Husayn had descendants through his son, Ali Zayn al-Abidin. This disproves those who claim that Al-Hasan and Al-Husayn had no offspring. Furthermore, the Mahdi is from Al-Hasan's progeny. The honor of lineage does not benefit its owner except through faith and righteous deeds, for "whoever is slowed down by his deeds will not be hastened by his lineage," and the most honorable among people in the sight of Allah are the most pious among them. Thus, honor is by piety, not by lineage. Attributing religious honor solely to lineage is among the rulings of the pre-Islamic era (Jahiliyyah). However, Islamic law has attached certain rulings to noble lineage, considering it a potential source of good. But if a man of noble lineage commits evil deeds, his offense is more severe and his punishment is harsher.