Modern Judaism is emphatically not the same "Judaism" of the time of Christ and not the same faith Christ praticed.
True Judaism is Orthodoxy!
So, yes, there is a continuation, but not of modern Rabbinic Judaism which was set up in opposition to Orthodoxy, but a continuation and fulfillment of the faith of Abraham, Issac, and Jacob.
Modern Orthodoxy is not either...it has evolved over time (more in its traditions than actual beliefs). Modern Rabbinic Judaism was instituted directly because there was no longer a Temple. Jewish Christians were a sect WITHIN Judaism from 30 AD to between 70 and 135 AD. there were these same problems even in Paul's day! Romans 11. Your type of love for Jews, even your Jewish brethren, will not "provoke them to jealousy", but provoke them to run from you.
13 For I speak to you Gentiles, inasmuch as I am the apostle of the Gentiles, I magnify mine office:
14 If by any means I may provoke to emulation them which are my flesh, and might save some of them.
15 For if the casting away of them be the reconciling of the world, what shall the receiving of them be, but life from the dead?
16 For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches.
17 And if some of the branches be broken off, and thou, being a wild olive tree, wert grafted in among them, and with them partakest of the root and fatness of the olive tree;
18 Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee.
19 Thou wilt say then, The branches were broken off, that I might be grafted in.
20 Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear:
21 For if God spared not the natural branches, take heed lest he also spare not thee.
22 Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off.
23 And they also, if they abide not still in unbelief, shall be grafted in: for God is able to graft them in again.
24 For if thou wert cut out of the olive tree which is wild by nature, and were grafted contrary to nature into a good olive tree: how much more shall these, which be the natural branches, be grafted into their own olive tree?