This topic come up in another thread. So I wanted to start a thread on it, to see what people think, and to see if an answer can be arrived at.
So, what are the origins of Christianity? Is it a continuation of Jewish revelation, the "new covenant" given to the Jews? Or is the origins of Christianity rooted in pagan myth?
This is an excellent question, one with no simple answer. Currently, the question is become much more complicated as well, as some new ideas are replacing old ones. But for a short answer, Christianity originated with Judaism.
A longer answer
To begin with, Christianity ultimately originates with Jesus. While Jesus did not intend to create a new religion, his actions and teaching eventually did. All of Jesus's first followers were also Jews. Paul, who spread helped spread the movement to gentiles, was also Jewish. The religion then has its basic roots in Judaism.
The original Jesus movement, starting after the death of Jesus, was also comprised of Jews. The leadership was in Jerusalem, and the leaders were Jewish. This remains true until at least after the time of Paul. According to Josephus, James (the brother of Jesus) died in about 62 C.E. James is recognized as the leader of the Jesus movement, so we can be quite sure that the leadership remained in Jerusalem until then. It could have continued there; however, by 70 C.E., with the destruction of the Temple and Jerusalem, it is very sure that the leadership moved, and it becomes more difficult to say who the leader of the movement is (most likely, there was no central leadership after the destruction of the Temple, and that the movement became shattered).
By this time, there are more gentiles coming into the fray. This would have been partially because of Paul, as well as several other missionaries who preached to gentiles. However, this would still have been within Judaism. There is general agreement that after the fall of the Temple in 70 C.E., Christianity was still within Judaism.
It is here that it gets quite complicated. While it is agreed that Christianity remained under Judaism for awhile after the destruction of the Temple, for how long is debated. One of the more traditional theories is that there was a severe rift between Christianity and what would become Rabbinical Judaism. By the second Jewish revolt, where many Christians supposedly refused to fight on behalf of the Jewish fighters, the rift had become so large that effectively there became two different religions. Much of this view is being changed though.
There is evidence that until at least the 4th century C.E., there remained Jewish-Christians. According to Daniel Boyarin, Christianity actually remained being a Jewish sect until then. Either way, Christianity remained quite connected to Judaism until the 4th century.
It is also generally accepted that Christianity exploded into many different sects. Some of them were extremely pagan. For instance, some of them did have a multitude of gods. What would become orthodox Christianity reacted to these other sects. Early Christian writers did heavily work within the Jewish tradition.
At the same time, from before the time of Jesus, Palestine and Judaism had been to a point Hellenized. They had been influenced by the Greek culture, and some areas were more influenced than others (Palestine would have been less influenced than the Diaspora, which means that Paul would have had been more influenced). So there would have been some influence of pagan ideas, and as time went on, and Christianity came into more contact with other pagan cultures, some aspects of pagan ideas would have filtered into it.