Does Jesus Pray For Everybody, Even Those Who Reject Him?

I can think of a lot of things in the Bible that would lead one to conclude that Jesus does pray for everyone.  Things like where it says that Jesus desires that all come to believe or something close to that. 
 

But if you look closer it doesn't really address Jesus praying.  The reason I bring this up is that I read the following verse today.

John 17:9(NKJV)

9“I pray for them. I do not pray for the world but for those whom You have given Me, for they are Yours. 

What caught my eye is that Jesus very specifically says that there are some He does not pray for.  He does not pray for the world.  I think that would surprise a lot of people.  To realize that Jesus Himself said that there are some who He doesn't pray for.

Who does He pray for?  Those whom God has given Him.  Who are those people?  Those who believe in Jesus.  Those who are saved.  Those who have chosen to become Children of God. 

Jesus goes on to say there is an additional group He prays for.

John 17:20-23(NKJV)

20“I do not pray for these alone, but also for those who £will believe in Me through their word; 

21that they all may be one, as You, Father, are in Me, and I in You; that they also may be one in Us, that the world may believe that You sent Me. 

22And the glory which You gave Me I have given them, that they may be one just as We are one: 

23I in them, and You in Me; that they may be made perfect in one, and that the world may know that You have sent Me, and have loved them as You have loved Me.

Those who will believe in the future.  The common denominator though is a belief in Jesus as the Son of God.  

Now why would Jesus be so mean as to exclude someone?  The answer is He isn't excluding anyone.  He has said that He is willing to pray for those who believe in Him.  Anyone is free to believe in Him or not believe in Him, that is their choice, Jesus isn't restricting them in any way.

Now here is what I find funny about those who are offended by this.  If you don't believe in Jesus to begin with, why would you care if He does or doesn't pray for you?  Hmm?