Long time ago, 9th grade I think, a friend of mine asked me why do Muslims believe in Prophet Muhammad (saw) if he's dead. Years later, back when I still had a facebook, I friended her and noticed she was quite Christiany, more than I ever noticed. In fact, a lot of my friends had this religious side they never really quite showed in school. Also, I have noticed from talking with many friends then and now that a lot of people have this one set understanding of religion and little to no knowledge about Islam passed we pray and fast a lot or something. They certainly have no idea why we believe what we believe as my friend's ignorance has shown.
Now my friend's ignorance was understandable but mine wasn't (to me anyways), I still regret the answer I gave which I suppose was partially a misunderstanding of the question in my confusion about why she thought something like that was important in believing in something. I said that in Islam, we believe prophets have Caliphs or Successors who succeed prophets and continue their message long after they have passed away. I guess that partly answers that question but it was lacking and weak overall. Back then I wasn't too interested in learning history and learning the details of Islam as I am now. Regardless, I had this implacable trust and belief in my faith and I studied it well after that, gaining knowledge over the years little by little.
The proper answer to my friend's question actually comes from moment in the history of Islam at the passing of Prophet Muhammad (saw). It was a severe shock for the Muslim community at the time and they were really unprepared for it b/c they had such loyalty and belief in him and they could not ever imagine that he'd ever leave them. It's natural after all, people don't consider for example when they're parents or close ones will die so when it happens, it sucks and a lot of people suffer severe depression.
Anyways, a close follower of his who would become the 2nd Caliph, Caliph Umar (ra) took up a sword and said that he didn't believe that Prophet Muhammad (saw) had died and if anyone said so, he would cut off his head. So then, the close friend of Prophet Muhammad (saw) who would soon be the 1st Caliph, Caliph Abu Bakr (ra), came to him and the congregation at the Prophet's (saw) mosque and told them 2 things which sobered everyone up. First, if anyone worshiped the Prophet (saw), he should know that he is dead but if anyone worshiped Allah, he should know that He is Everlasting. Second, he proceeded to quote the following verse from the Qur'an:
[3:145] And Muhammad is only a Messenger. Verily, all Messengers have passed away before him. If then he die or be slain, will you turn back on your heels? And he who turns back on his heels shall not harm Allah at all. And Allah will certainly reward the grateful.
The truth in our eyes is submission to Allah and the message he sent to us through His messengers. Just b/c His messengers have passed on doesn't mean His message will disappear. This is a mistake tribes in the time of Caliph Abu Bakr (ra) made that were far away from Medina, the city of the Prophet (saw), who still had not received proper education from Islam and some of them treated it as another pact with some leader and were therefore persuaded by local troublemakers to raise arms against Medina and the Muslims and not pay any Zakat or the tax on Muslims for the benefit of the state and society.
In our belief, Prophet Jesus (as) was a man and not God or His son as Christians believe. They also believed that no prophet of God has ever been fully innocent of a crime or indecency, to my knowledge anyways, and you pick any prophet, you'll find there is some moment of weakness in their life given about them in the Old Testament going all the way back to Prophet Adam (as). We do not believe in the truth of these accusations and of course the Qur'an, which is the book for all mankind, denounces the accusations and negative accounts given of them.
As a Muslim, we have a greatly differing view from both beliefs and much more so about Prophet Muhammad (saw) as we see him as the chief of all Prophets. We believe God to be the ultimate power to whom we hold allegiance and Prophet is a messenger as any messenger or diplomat is messenger to any head of state and as a messenger, he would be treated with similar reverence and respect as would be given the King or President he represents if he were in the presence of whoever was greeting him. Furthermore, the diplomat would do his best to emulate the outstanding qualities of his King and so would not be the shortfalls people would try to ascribe to him. Now if this diplomat is killed, God forbid, then the King or President would live on and maintain His power.
Another very important verse is the following which is usually recite when someone passes away or something important is lost but also can be said when something causing grief happens:
[2:147] Who, when a misfortune overtakes them, say, ‘Surely, to Allah we belong and to Him shall we return.’
It is true, our Prophet Muhammad (saw) has passed away but so will we all as have all Prophets and peoples before us but that makes their message no less true or credible or good. In fact, I feel closer to the Prophet (saw) and all prophets b/c of their humanity and their closeness to God and if I believed in their truth, I would be rewarded similar to them in time, Inshallah.