When Brian grabs the first bottle of liquor he takes one swig before continuing the conversation with Anna. When the next shot of him is shown less than 5 seconds later the bottle is less than half full.
During the conversation between Jake and Brian, after the fight in the synagogue, the camera is on Jake's face from behind Brian's shoulder at one point. You can clearly see that he is not talking, and yet we hear Brian's voice.
Brian shows a picture to the bartender, clearly showing the kids' torsos. Then when Brian starts explaining who the kids in the picture are and we see the picture again, the kids are much closer to the camera and their torsos are no longer visible.
During Jake's and Brian's slow-motion walk down the busy street, there is a man wearing glasses and a cream-colored jacket. As they approach, the man repeatedly jumps in front of/behind them between shots.
After Brian kisses Anna he goes to the liquor cabinet, swigs the first bottle and puts the second in his jacket. A few moments later, he takes another swig of the first and puts the second in his pocket again.
When Anna and Jake's mother sit on the sofa looking at pictures of the Schram's wedding, you can see that Anna's hair line is darker the rest of her hair - she obviously has to redye. In the next scene which plays maybe 1 hour later in the story, her hair is all over blonde.
The rabbi works with a young man on his Torah portion but refers to it as a haftarah. Also, the bar mitzvah boy is studying the first chapter of Genesis and has his bar mitzvah before the Jewish New Year. This portion is read after the Jewish New Year, not before.
Some of the scenes that supposedly take place on Yom Kippur on not realistic. Yom Kippur is a solemn day of fasting, prayer and repentance, and observant Jews spend basically the entire day in temple services...following Jake's Yom Kippur "speech" as they called it (there are no "speeches," they are called sermons) the board of Jake's temple appears to have a meeting to discuss his career, this would/could never happen since no work is allowed on that day. Also it appears that right after that meeting concludes and Jake finds out he still has a job he then gets Anna and goes to the karaoke party....there is no way Jake, Rabbi Lewis and Jake's mother would be at a karaoke party on Yom Kippur.
After nearly catching Anna in Jake's apartment, Brian is in a rush to get to the karaoke store because "it closes in like 20 minutes", but Anna is on her way to work suggesting it's sometime in the early morning. It's doubtful a store would close at such an early hour.