The questions of "where do you mean in the Levant" and "what does living continuously mean" arise with this question.

The Levant is a large and diverse region, in which Jews have lived in various communities and migrated through over the past 2000 years. Jerusalem, the Galilee, and Jaffa (among other cities/regions) in historic Palestine/the modern state of Israel have had continuous Jewish presence since the Roman expulsion, in addition to waves of religious, economic, and political migration of Jews from throughout the Mediterranean, Iraq, central and Eastern Europe, and more. What is today Lebanon has likely had Jewish communities since the end of the Jewish revolts against the Romans (Bar Kochva Revolt in the second century), and saw the establishment of major Jewish centers of study in the middle ages. Beirut in particular had both long-standing communities and was host to increased Jewish migration in the late 19th and early 20th c. from other places in the Ottoman Empire. Finally, Beyazid II famously declared the Ottoman Empire would welcome Jews expelled from Spain at the end of the 15th c., so many Sephardic Jews ended up in communities in what is now Israel, Syria, and Lebanon (Damascus, Beirut, Safed, Jaffa, Jerusalem, more).