A common saw each Christmas is that Christianity is a warmed-over paganism.  The theory goes that Jesus was, variously, a dead rabbi with a girlfriend, or never existed, but that Christians, for some reason, thought it would be a good a idea to stop calling Osiris or some other dying and rising god of fertility myth by his proper name, start calling him by the name of Jesus of Nazareth instead, and begin worshipping him instead of just sticking with the old god.  All of this is proven, we are told, by the fact that December  25 was the Feast of the Unconquered Sun.  So there you are.

Here's the thing though.  The Feast of the Unconquered Sun was not made a feast until 274.  Meanwhile, we have records from decades before this, showing that the Christians were already celebrating the birthday of Jesus "eight days before the Kalends of January" according to St. Hippolytus.  In other words, on December 25.  The Feast of the Unconquered Sun turns out, in fact, to be a pagan attempt to co-opt Christmas, not the other way round.The real reason Christmas is on December 25 has nothing to do with the winter solstice for the very good reason that, despite modern myth, early Christians had no interest in trying to worship pagan gods.

On the contrary the complete lack of interest (and not infrequent contempt for) pagan rites, stories, and holy books is what strikes you in the face when you actually read ancient Christian sources.  Again and again, Christians tell their audiences not, "Hey!  You like Osiris?  So do we!  Only we call him "Jesus" cuz Hebrew names are cooler!" but "Turn away from these vain things" and even "The gods of the Gentiles are demons."  To be sure, they also grant that God has not left himself without witness among the Gentiles, but the tone is very much "Even these clueless Gentiles have a tiny inkling of the truth, which the revelation of the one true God of Israel through his Son Jesus Christ can correct and complete."  It is not at all, "Let's import some Apollo and Isis worship into our preaching to spice things up for Gentiles."  Indeed, when the Christmas sceptic is not busy damning the Church for being nothing but pagan myth, he often forgets himself and damns the Church for being so bigoted and closed to paganism.

The source of the confusion is this: The Church took no interest in pagan traditions about their gods.  They cared only about Jewish and Christian scripture and tradition.  That's it.  That's all.  They didn't give a fig about Greek or Roman gods.  They did not care about who worship was done at the Temple of Diana in Ephesus.  But they cared passionately about how it was done in the Temple of God in Jerusalem.  

And in reckoning the date of Jesus' birth they did likewise.  Pagan solstice festivals were complete unimportant to them, but they cared intensely about a Jewish tradition (unattested in Scripture but quite real in Jewish (and spilling over into Christian) culture concerning something called "integral age". The idea was that a true prophet's death day would be the same as the day he was conceived or born.

We see this tradition attested by none other than <a href="http://www.logoslibrary.org/augustine/trinity/0405.html">St. Augustine in his book on the Trinity</a>: "He is believed to have been conceived on the 25th of March, upon which day also he suffered; so the womb of the Virgin, in which he was conceived, where no one of mortals was begotten, corresponds to the new grave in which he was buried, wherein was never man laid, neither before him nor since." Nine months after March 25=December 25.

And remember, it doesn't matter if you think the idea of integral age is rubbish. What matters is that this, not the solstice and some need for warmed-over paganism, was what drove the thinking of the early Church. The early Church simply did not care about imitating paganism cultic practice, stories or doctrines. Their focus was on Jewish and Christian scripture and tradition and nothing else. Sure they sometimes put Christian content into pagan forms (Easter eggs, Christmas trees, wedding rings...). But they never put pagan content into Christian forms.

"But what about the very name 'Easter'? It comes from the pagan goddess 'Eostre'!"

Right you are. And 'Friday' comes from "Freya's Day". Does that make you a Norse pagan who worships Freya when you say, "Thank God it's Friday?" No it makes you an inheritor of a Germanic language group that didn't bother to change the names of some of its days. But in all Latin language groups (and beyond) what we English speakers call "Easter" is known by some variation on Pascha, or (as it's translated into English) "Passover". Why? Because Jesus was crucified on the eve of Passover and is our Passover sacrifice. Because early Christians only cared about Jewish and Christian scripture and tradition.  And that tradition is about an actual historical human being who was crucified, not in cloud cuckoo land, Olympus, Valhalla or some other mythic realm, but in Jerusalem under a workaday Roman procurator who has <a href="https://en.wikipedia.org/wiki/Pilate_stone">left behind archeological evidence of his existence.</a>

The notion that Jesus never existed, like the notion that his passion and resurrection are warmed-over pagan myths with no foundation in history and eyewitness testimony, is something that only a particularly credulous conspiracy theorist could buy.