That is one of the easier questions I have had.  Jesus is the light of the world (John 8:12).  Jesus proved this in so many ways, it is difficult to count.  Almost immediately after telling the Jews in Jerusalem that he is the light of the world, Jesus healed a man who had been born blind.  What better way to confirm Jesus’ claim to be the light of the world?  In fact, Jesus healed dozens, certainly and more likely hundreds of people of blindness.  Not only did the one who said, “I am the light of the world” (also in John 12:36), heal the blind, he walked on water, turned water to wine, healed the lame, calmed a raging storm, raised Jarius’ daughter and Lazarus from the dead, and was himself raised from the dead.  Jesus lived a sinless life.  Jesus proved he was the light of the world by the miracles he worked.

 As you point out, Muhammad made a similar claim to that of Jesus.  Actually, his claim was not as strong, but it was similar–that he was a lamp, rather than the light of the world.  Muhammad claimed to be a lamp, whereas Jesus claimed to be the light of the entire world.  Having said that, where did Muhammad get the authority to make such a claim?  What public miracles did he work to validate his claim?  The answer is that he worked no public miracles.  What prophecy did he fulfill?  None that I know of.  Where did Muhammad get the authority to claim to be  the lamp?  I am not sure, but Jesus has all authority in heaven and on earth (Matthew 28:20), and he proved his authority again and again, through his fulfillment of messianic prophecy, through his miracles, and through his sinless life.

This was a relatively easy question to answer.  Jesus is THE light.  Muhammad claimed to be A light, but provide no strong evidence to back up his claim.

John Oakes