Is the Old Testament a Jewish book? Is the Old Testament a Jewish religous book and should it be discarded by Christians today? If not what does it add to Christians today?
You are asking two completely separate questions.

1. Is the Old Testament a Jewish book?

Insofar as the Torah was revealed first to the Jews, yes. As Romans 9 points out


  Theirs [The Jews] is the adoption to sonship; theirs the divine glory, the covenants, the receiving of the law, the temple worship and the promises. 5 Theirs are the patriarchs, and from them is traced the human ancestry of the Messiah, who is God over all, forever praised


But, is the promise made to Jews alone?  Paul continues:


  It is not as though God’s word had failed. For not all who are descended from Israel are Israel. 7 Nor because they are his descendants are they all Abraham’s children. On the contrary, “It is through Isaac that your offspring will be reckoned.”[b] 8 In other words, it is not the children by physical descent who are God’s children, but it is the children of the promise who are regarded as Abraham’s offspring.


Whether or not we are descended from Abraham (or even Isaac) here is irrelevant - Paul admits it is not the children by physical descent who are God's children. Whilest I am always a little bit leery of putting the church in the place of the Jews in all things, it is fair to say that we are heirs of the same promise.

2. Should it be discarded?

Jesus himself said in the Sermon on the Mount (Matt 5:17)


  "Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them.


Futhermore, Paul - the biggest opponent of "the law" if there ever was one - even he says (Galatians 3:17):


  17 What I mean is this: The law, introduced 430 years later, does not set aside the covenant previously established by God and thus do away with the promise. 18 For if the inheritance depends on the law, then it no longer depends on the promise; but God in his grace gave it to Abraham through a promise.


He goes on to call the law a "pedagogica" - think of it as a nanny - that leads us to Christ. It is the thing that makes us aware of our sin

In Romans 7, he makes the same point:


  Nevertheless, I would not have known what sin was had it not been for the law. For I would not have known what coveting really was if the law had not said, “You shall not covet.”[b] 8 But sin, seizing the opportunity afforded by the commandment, produced in me every kind of coveting. For apart from the law, sin was dead. 9 Once I was alive apart from the law; but when the commandment came, sin sprang to life and I died. 10 I found that the very commandment that was intended to bring life actually brought death. 11 For sin, seizing the opportunity afforded by the commandment, deceived me, and through the commandment put me to death. 12 So then, the law is holy, and the commandment is holy, righteous and good.


In all these things, the OT law still has a purpose - it makes us aware of our need for God.

Frankly, it also tells us who God is.

So, in short, no, we can't discard the OT - it's the foundation without which grace makes no sense.