Something I often stress to students of early Christianity is that this Jesus movement was very much a form of Judaism in its origins. The peasant Jesus was a Jew, and all the earliest followers of Jesus were Jews, Jews who continued to feel that following the law (the Torah) was humanity’s response to God’s covenant with his people (Paul, the Jewish Pharisee, was a bit of an exception in not requiring that gentiles follow certain aspects of the Jewish law — especially circumcision and food laws — in order to join. Still Paul was very much a Jew and did not object to Jewish followers of Jesus following the law and, in some respects, expected gentiles to follow other aspects of the law beyond those that created a social or status distinction).
So just about every portrayal of Jesus in the first century would naturally reflect the Jewishness of Jesus, as is in indeed the case. However, among the gospels in the New Testament, the Gospel of Matthew stresses perhaps more than others the Jewishness of Jesus. Jesus is presented as the ultimate fulfillment of the Jewish scriptures in over a dozen fulfillment citations. The gospel begins by presenting Jesus as the son of David, the anointed king par excellence.
Furthermore, Jesus is often presented as the new Moses, as in the birth narrative. This continuing theme of Jesus as the expected prophet like Moses continues in what Matthew has as the sermon on the mount (Matthew 5-7). There Matthew’s Jesus affirms the continuing validity of the Jewish law for the followers of Jesus:
“Think not that I have come to abolish the law and the prophets; I have come not to abolish them but to fulfil them. For truly, I say to you, till heaven and earth pass away, not an iota, not a dot, will pass from the law until all is accomplished. Whoever then relaxes one of the least of these commandments and teaches men so, shall be called least in the kingdom of heaven; but he who does them and teaches them shall be called great in the kingdom of heaven. For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven” (Matthew 5:17-20)
Jesus’ followers were to follow the law in a way that excelled the Pharisees, who had a reputation (outside of the gospels and in this passage in Matthew) for carefully following the law in their daily lives, beyond the practice of many other Jews. This is the interpretive key to the material that follows (falsely called “antitheses”) which have Jesus quoting the law of Moses and interpreting it in a way that strongly affirms the original intention of the laws (as Jesus’ Matthew understood it). These are not replacements for the law of Moses, but rather a radicalization of the reason why those laws were given by God, in Matthew’s view. Matthew and some in the community for which his gospel was written in the late first century evidently continued to place an importance on following the law and continued to think of themselves as Jews in this way.