In the opening scene at a swimming pool, the narrator's voice (Woody Allen's) says that it took place 'in the late thirties'. Yet, towards the beginning of the movie, soon after arriving in Hollywood, Bobby comes out of a movie theater advertising on billboards and posters a Barbara Stanwyck film "The Woman in Red" _ which was released in 1935.
During the nightclub sequences, apart from the New Years Eve scene when they play "Auld Lang Syne", the band musicians are obviously miming or not playing their instruments, although we hear them play various jazz tunes.
Twice, one of the characters states that Judaism does not believe in an afterlife. This is untrue. Judaism does indeed recognize an afterlife. Many books have been written on the subject including "Does the Soul Survive? A Jewish Journey to Belief in Afterlife, Past Lives & Living with Purpose" by Rabbi Elie Kaplan Spitz (2015),