I covered a lot of ground in my reading of Hans Dieter Betz's commentary on The Sermon on the Mount. I'll roughly summarize some of what I read in this post.
In Matthew 5:17-19, Jesus says (in the KJV): "Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven."
This passage was emphasized when I was growing up in Armstrongism, a denomination that believes that Christians have to observe the Old Testament law. But many evangelicals have responded that this passage is not saying that believers have to observe the Torah, but rather that Jesus fulfilled the Torah, which means that we do not have to observe all of it literally (i.e., the Sabbath, the sacrifices, etc.). Moreover, when Jesus talks about keeping the least of the commandments, they argue, Jesus is speaking of his own commandments in the Sermon on the Mount, not the laws in the Torah.
Betz disagrees with that particular evangelical spiel. He agrees that "these commandments" concerns Jesus' commands, but, unlike the evangelicals, he does not maintain that Jesus is in the Sermon on the Mount coming up with a new law to replace or to fulfill the old law. Rather, Betz holds that Jesus is interpreting the Torah. And Betz does not think that Jesus is making commands as part of his Messianic authority, but rather that Jesus is simply exercising authority as an interpreter of the Torah. For Betz, Jesus in the Sermon on the Mount regards the Torah as normative as long as this transient world exists. This Jesus, according to Betz, differs from Paul, who in Galatians believed that the Torah was temporary and would only last until the work of Christ. When Jesus refers to the "least in the kingdom of heaven", Betz believes that is a reference to Paul, who called himself the least of the apostles in I Corinthians 15:9. According to Matthew 5:19, Betz states, Paul would still enter the kingdom, but he would be the least within it because of his laxity regarding the Torah.
Betz does not believe that the Sermon on the Mount is by the author of the Gospel of Matthew, but he regards it as pre-Matthean. He provides reasons for this, and he'll probably offer more as I continue reading: The Sermon on the Mount lacks discussion of Christology and Jesus' suffering, the Sermon is not as apocalyptic as Matthew (though Betz acknowledges that Matthew 5:17-19's reference to the passing of the earth is rather apocalyptic), etc. But I think that the Sermon on the Mount may be from the author of Matthew, and that it overlaps with Matthean themes. For example, the Sermon on the Mount says that the Torah is still authoritative. In the Gospel of Matthew, there is an attempt to legally justify Jesus' actions on the basis of the Torah. For example, in Matthew 12, Jesus affirms that the priests profane the Sabbath when they offer sacrifices on the Sabbath, yet they are guiltless, and that he is greater than the priests, which means that he can legally profane the Sabbath, at times. Come to think of it, that does appear to nullify the Sabbath, on some level. But Jesus is trying to show that he can lawfully do so, using legal argumentation that the Pharisees employed. For Matthew's Jesus, the law is not void, and so he has to justify his actions with reference to the law.
That brings me to my next point: Betz distinguishes between the law and justice, as he draws from such thinkers as Aristotle, whose thought Betz believes had some impact on the one who wrote the Sermon on the Mount (even if he didn't know that he was drawing from Aristotle specifically). Jesus fulfills the law when he uses it to further justice, Betz claims, which means that the law can be technically observed and yet not further justice (something even some of the rabbis thought, according to Betz). I think that one implication of this is that, for Jesus in the Sermon on the Mount, the outward observance of the law is not sufficient, which is why Jesus says that hate is murder and lust is adultery. Whether Betz believes that Jesus is making the law all about love for neighbor and thus is nullifying rituals, I do not know. I remember that a professor told me that Betz believes this, but, if that were the case, what would be the Sermon on the Mount's beef with Paul, who also nullified ritual while saying that love for neighbor was authoritative?
Betz's discussion of Matthew 5:28, in which Jesus says that lusting after a woman is adultery, was interesting. Betz argues that Jesus was criticizing looking upon a married woman, for that could lead to adultery. Betz looks at ancient passages in which the eye is the first step on the path towards adultery. Jerome, however, translated Matthew 5:28 to mean that men looking at any adult female (not just another man's wife) is adulterous----but Betz maintains that Jerome meant any woman other than one's own wife.
On the succeeding verses, which concern cutting off one's right hand, Betz regards that as hyperbole, and he cites ancient parallels in which moral seriousness was expressed in terms of mutilating limbs.