Jesus is the fulfillment of the Old Testament. The promises and ritual of the Old Testament point toward Jesus' life and work.
God's covenant and promises with Israel were fulfilled in the final form when his only son Jesus took on human flesh. The Old Testament points toward the final victory of Jesus.
The continuity between the Old and New Testament is that the Old Testament finds its ultimate significance in the person and work of Jesus. Often the discontinuity of the testaments is exaggerated.
In reality the Old Testament is in complete continuity with the New. It is unclear how easily much of the prophecy in the Old Testament would have been understood.
Liberal scholars often attempt to make a case that since the prophecy in the Old Testament might have at times been misunderstood by its readers the prophecy could not be about Jesus. It is of course a fallacious argument of the worst type. The misunderstanding of a text does not make the misunderstanding the valid reading of a text.
The Old Testament points to Jesus. The final covenant between God and mankind commenced when God sent his only son Jesus to fulfill all righteousness on our behalf.