Historically speaking, and theologically speaking, it is more correct to say that Judaism and Christianity both find their roots in the Old Testament faith. But Judaism is a religion based on the traditions of the elders which nullifies that faith. Whereas, Christianity is a religion based on the belief that Jesus is the fulfilment of the Old Testament promise for the Messiah. Hence, to say that Christianity came out of Judaism is incorrect. Christianity is the hope of Israel because Christ is the hope of Israel.
This is an easy mistake to make. We often think anachronistically. What Jewish people believe we have called Judaism for nearly two millennia now, so it is natural to confuse this with the faith of the Old Testament. However, this is not what the New Testament teaches. Yes, Jesus is an Israelite, a Jewish man of the tribe of Judah, a Judean of Judeans, descended through David according to the flesh and born of Mary, he is the God Man who sits enthroned in heaven, having defeated sin, death and the devil, and having fulfilled the law, the Son of God, God the Son. There can be no doubt that Christianity is connected to the ancient faith of Israel, and anyone who denies this is just wrong. Christianity is the fulfilment of the hope of Israel. But Judaism finds its origin in a rejection of the authority of the law of Moses. So to say Christianity came out of Judaism is simply incorrect.
We should train ourselves to think on these issues like the Bible does. When we do so this also helps clear up confusion about who the people of God are. As Jesus himself said,
“46 While he was still speaking to the people, behold, his mother and his brothers stood outside, asking to speak to him. 48 But he replied to the man who told him, “Who is my mother, and who are my brothers?” 49 And stretching out his hand toward his disciples, he said, “Here are my mother and my brothers! 50 For whoever does the will of my Father in heaven is my brother and sister and mother” (Matt. 12:46-50).
Only those who do the will of the Father in heaven are God’s people, or God’s family. And the heavenly Father’s will is that we should listen to and believe in Jesus, ““This is my beloved Son, with whom I am well pleased; listen to him” (Matt. 17:5). The Pharisees - generally speaking as some repented of this - were the first and most prominent group to oppose Jesus. We are not in their tradition. And we should make sure that people understand that. Christianity is the fulfilment of the law, because Christ is the fulfilment of the law.