If you set up that kind of a false dichotomy -- that the disciples left Jesus at the end, and thus they were not believers here -- then none of the teachings of Jesus were given to believers. In the mind of Jesus, the disciples were believers in training. These were those which he had personally selected and through whom he was going to build and establish his church. In 5:13, Jesus says, "You are the salt of the earth." In 14, he says, "You are the light of the world." Verse 16, "Let your light shine before men, that they may see your good works, and glorify your Father in heaven."
Verse 11 goes in here also, "Blessed are you when they revile you and say all manner of evil against you falsely for my sake. Rejoice and be exceedingly glad, for so persecuted they the prophets who were before you." Here Jesus forms a dichotomy between the disciples and "they." The same "they" are those who persecuted the prophets. This "they" -- we know from Matthew 23 -- would include the unbelieving Jews who persecuted the prophets; and contextually, it specifically included the Pharisees, who are said to be the murderers of the prophets, as Jesus charges that all of the blood of all of the prophets would be charged against that unbelieving generation.
But in Jesus' statement of verse 11, he is also linking the disciples up with the prophets. The disciples would enjoy the same persecution which had previously come to the prophets. The prophets were the messengers of God who existed before you; Jesus is saying the you are the messengers of God in the now.
Jesus is, in the Sermon on the Mount, laying the foundation for the new morality of the kingdom of God, with an internal righteousness which will exceed the external righteousness of the Pharisees (5:20). It will be the reconciled kingdom who desires to be reconciled to their brothers (5:23-24). It will be the kingdom the eschews lust (27-30), and that upholds the covenant of marriage (31-32). It will be the kingdom that lives by its word, that does not return evil for evil, that loves and gives to even those who are abusive, and who does good to those who hate them, and prays for those who despitefully use them.
In this way, I believe, we become perfect even as our Father in heaven in perfect. We become a people that reflects the righteousness of God. We have Christ's image contained in vessels of clay (2 Corinthians 4:7ff) and we reflect and we image the very glory of God. And, by the way, the Sermon on the Mount does not end here, but it continues on through chapter 6, and does not end until 7:27. In these chapters we find many instructions given to believers, such as how to pray, how to give discreetly, how to fast, how to lay up treasures in heaven, etc. There is the constant dichotomy here again between us and the hypocrites -- and there is constant reference to the fact that God is our Father in heaven. So, no, I have to disagree: the Sermon is written to Christians; and its standards are not unreachable, but are the reflect the life of the called out, redeemed and sanctified communion.