Is the sermon on the mount intended for unbelievers?

As I read your question, it seems to be asking more about what were the attitudes of his audience than whether or not they were believers since this was well before Jesus death and resurrection.

So if that is the case we need to look at the circumstances when Jesus was ministering to the Jews.

Some of the history of the Jews;

Having been brought out of Slavery in Egypt the Hebrews had become a mighty nation under such leaders as King Saul, King David and King Solomon. But they had abandoned God in favor of serving other gods and God had for centuries saved the nation, only to have them disobey again and again.

God had told the Nation of Israel that he would protect them as long as they served him, but would abandon them if they strayed.

All Scripture is quoted from the King James translation, unless otherwise noted.

Leviticus 26:36 through 39 And upon them that are left alive of you I will send a faintness into their hearts in the lands of their enemies; and the sound of a shaken leaf shall chase them; and they shall flee, as fleeing from a sword; and they shall fall when none pursueth. And they shall fall one upon another, as it were before a sword, when none pursueth: and ye shall have no power to stand before your enemies. And ye shall perish among the heathen, and the land of your enemies shall eat you up. And they that are left of you shall pine away in their iniquity in your enemies' lands; and also in the iniquities of their fathers shall they pine away with them.

At the time Jesus came to reclaim the Nation of Israel:

Matthew 15:24 But he answered and said, I am not sent but unto the lost sheep of the house of Israel.

They had once again rejected God and were under harsh rule by the Roman Empire, and the Jews, were and still are looking for a Messiah. Only their conception of a Messiah is someone like a David who will come and restore them to their former greatness.

So this was the attitude of his audience, and not only did the Apostles expect him to physically save the nation as those Kings had done, but that was the general belief of all he was ministering to, It never occurred to them that Jesus might not lead them in battle and be talking about a spiritual Kingdom.

It was only after his death and even more his resurrection to live again among them that the miracles and teaching were understood to be of a spiritual nature, and even unto this day the empty tomb is the basis for Christianity.

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.