Yes, more or less. This is all of course quite controversial, both for current political reasons and for scholarly reasons. As a historian, I'll take a few moments and deal with the latter. The big problem that we have as schoalrs is that we don't really know a ton about the composition of the Quran. Lots of people have theories, some more plausible than others. Some theories are even backed by evidence! But the thing is, everyone knows that Muhammad didn't write down the Quran. It was compiled in an authoritative version some time beginning decades after his death, as Islam really formed into a coherent religion and the Umayyads became rulers of a real empire. So traditions were collected, especially from people who had been companions of the prophet, and the text was assembled. Later, a similar process occurred with the Hadith, the name for the deeds of the Prophet (stories about things he did) that provide much of the context for the Quran and our understanding of life in Arabia during the prophetic era. (There are a couple contemporary texts, like a treaty that was signed between Muhammad and a local Jewish community, and a couple Byzantine texts that mention him--we'll get to those when we get to them). But basically you can think that these collections of texts are accurate, or not. Some people are total skeptics and basically think that we can't make any real historical claims based on these texts. On the other side we have people who think that everything is true and literally the word of God sent down by the archangel Gabriel etc etc. I fall somewhere in between and geenrally think that we can use these texts to situate the early Muslim movement within its late antique context.

They key to the late antique context of early Islam is that it was highly charged and eschatological. The Arabian Peninsula had gotten swept up into the genration long war to end all wars between eastern Rome (Byzantium) and Persia, which lasted from 602-628 and ended with the recovery of the relic of the true cross that had been taken by the Persians from Jerusalem. When it was brought back to the Holy City many people thought that the world was literally going to end. This is the context in which Muhammad is preaching, and he leaves Medina for Mecca in 632, marking the beginning of the Islamic conquests.

Fred Donner has argued persuasively that this movement did not see itself as a "religion" per se at the beginning, but rather was a radically fundamentalist monotheistic movement known as the "believers," which preached total monotheism and the coming end of the world, which required repentance. Thus, in this sense, we can see Muhammad as the last prophet, an eschatological figure something like the Jewish Messiah or the early Christian Son of Man, who had come to deliver a message and usher in a final age of warfare before the establishment of a righteous kingdom on earth, see, e.g., MAthew 16:27-28, " For the Son of man shall come in the glory of his Father with his angels; and then he shall reward every man according to his works. Verily I say unto you, There be some standing here, which shall not taste of death, till they see the Son of man coming in his kingdom."

So basically, to sum up, according to the most careful and cautious modern scholarly theory, we see Muhammad as the representative of a radically puritanical movement that welcomed all monotheists to reform their lives before the end time. This emerged in a context of generational war that had shattered the empires of the east and brought eschatological thinking into the minds of everyone. Similar movement can be seen in Christianity and Judaism at the time.

Fred Donner, Muhammad and the Believers: At the Origins of Islam

Patricia Crone. Meccan Trade and the Rise of Islam. Princeton University Press. 1987

Steven Shoemaker, The Death of a Prophet: The End of Muhammad’s Life and the Beginnings of Islam

I think it's worth clarifying that Islam isn't quite a continuation of the Old and New Testaments, the exact quotes escape me but you'll find several mentions of the specific scriptures being continued within the Qur'an itself as "To Moses we gave the Torah, to Jesus we gave the Gospel and to David we gave the Psalms". Essentially those are the "accepted" scriptures of Judaism and Christianity, the rest is largely not.

Follow up,

I figure there must have been many attempts of claimed Messiahs besides Jesus and mohammed, do we have examples of these failed prophets and why did they not succeeded?

Perhaps I've misunderstood your question, but it seems some of the answers are claiming the Prophet of Islam himself is considered messiah. If that's the case, that is false. Jesus is considered the islamic messiah.

From the Qur'an (3:45):

Behold! the angels said: "O Mary! Allah giveth thee glad tidings of a Word from Him: his name will be Christ Jesus, the son of Mary, held in honour in this world and the Hereafter and of (the company of) those nearest to Allah;

From the Qur'an (4:157):

That they said (in boast), "We killed Christ Jesus the son of Mary, the Messenger of Allah";- but they killed him not, nor crucified him, but so it was made to appear to them, and those who differ therein are full of doubts, with no (certain) knowledge, but only conjecture to follow, for of a surety they killed him not:-

From the Qur'an (4:171):

O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity": desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs.

Mohammed was a prophet of God

did you mean to make that claim as a definitive statement?