Theological continuity would mean that Islam accepts and extends, ie that Jesus is the anticipated Messiah subsequently crucified and risen, not to mention accepting Jesus as the son of God and final prophet.
I have yet to find any Islamic faction that holds that to be true :)
Christianity has a high degree of continuity with Judaism, but rejects important aspects of Jewish theology, such as Kosher and Sabbath-keeping. So while Christianity accepts large amounts of Jewish theology, and extends new ideas, it doesn't universally accept everything that came before it.
Similarly, Islam accepts large amounts of Christian theology (and indeed reintroduces some Jewish practice) and brings new beliefs with it, but doesn't bring every tenet along.
Admittedly Jesus' divinity is a very big part of Christianity to reject; but then, the idea of the trinity would be an equally big pill to swallow for a Jew ("hear, O Israel: the LORD your God is one").
https://nyuscholars.nyu.edu/files/42476117/Abrogated_Rulings...