A common assumption concerning what Jesus said in the Sermon of the Mount is to consider the prerogatives of Christ's teachings as discordant (to greater and sometimes lesser degrees) with the Jewish/Pharisaic values and teachings of the religious leaders of Jesus' day. Thanks to the insight of late scholars like David Flusser and Shmuel Safrai, a wrench has been thrown into this oversimplified view of Pharisaism. In fact, a robust ethic is found in the language of Rabbinical writings that not only echoes much of the same insights found in Christ's Sermon on the Mount, but also the philosophical framework underlying it. In fact, there is even in Rabbinic ethics a precise terminology to characterize Christ's exhortation to "exceed the righteousness" of those who teach the injunctions of the Law (Matthew 5:20). It is called "going beyond the line of Torah". To go "beyond the line" means to observe the ethical trust of the Torah (and even innovate or improve) in matters where the Torah remains silent.
I offer here below a diagram which I hope begins to show the distinction between "Torah Observance", in the strict sense, and what is called (among other things) the "Paths of the Righteous" to describe Jewish observance which goes beyond the letter of the Law. To read further on this concept, I encourage reading Chapter XIII "The Commandments" of E.E. Urbach's The Sages and Chapter VI "The Two Ways and the Sermon of the Mount" of Huub Van de Sandt and David Flusser's The Didache: Its Jewish Sources and Its Place in Early Judaism and Christianity.